Understand that the probability of either of two disjoint events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities.

Understand that probability of a combination of two separate, disjoint, events can be found by adding up the individual probabilities. Understand that mutually exclusive, or disjoint, events are two events that cannot happen at the same time. Disjoint events are based on events happening in the same trial. Know how to find the sum of two separate probabilities to name a unified probability. Understand that when independent trials are conducted, one event does not affect the outcome of the second, same event. Know how to find the probability of two independent events by multiplying the probabilities of the separate events. Independent events are based on two trials or more. Identify disjoint events and independent events. Calculate the probability of two disjoint events by adding the probabilities. Calculate the probability of two independent events by multiplying the probabilities.

There are six cards with the numbers 1 – 6 on each card. One card is drawn at a time.

A) A number less than 4 and even is drawn.

B) A number less than three is drawn and a number greater than 4 is drawn.

(The events in B are disjoint, because these events cannot happen at the same time.)

(2)

Identify the independent events.

There are six cards with the numbers 1 – 6 on each card.

One card is drawn, a 5 and the card is put back into the pile. The cards are shuffled. A card is chosen again, it is 7.

One card is drawn, a 4. The 4 is placed in the trash. A second card is drawn, a 2.

(The events in A are independent, because the first outcome did not affect the second outcome when the card was replaced.)

(3)

There are six cards with the numbers 1 – 6 on each card. What is the probability that you will draw a prime number or a multiple of 6? Remember, if these events cannot happen at the same time, they are disjoint. Identify the kind of events described. Explain.

(4/6 or 67%, disjoint events, same trial)

(4)

You are conducting an experiment. There are six cards with the numbers 1 – 6 on each card. What is the probability that you will roll a 4 on your first try and a 6 on your second try? Identify the kind of events described. Explain.

(1/36 or about 2.7%, independent events, two trials)

(5)

Determine if the event is disjoint. Explain.

There are six cards with the numbers 1 – 6 on each card. What is the probability that you will pull an even number or a multiple of 3?

(These events are not disjoint, they have some overlapping favorable outcomes. For example, 6 is both even and a multiple of 3.)

Disjoint events are a pair events that cannot happen at the same time or on the same trial. These events are also, sometimes called mutually exclusive events. A trial is one experiment. That is one card drawn, spinner spun, cube rolled, etc… For example, if you put alphabet cards in a box, it would be impossible for you to draw one card with a vowel and a consonant on it. This is because vowels are not consonants and consonants are never vowels. When two events have no relation, never overlap, or are opposites, it will be impossible to pair those events so they happen at the same time. They will have no favorable outcomes, or possibilities for an event to occur, that are the same or overlapping. This would hold true for odds, evens, etc… You cannot find a number that is both odd and even. There are many other possibilities. Each even must be thought out thoroughly, with notes drawn for best success at looking for overlapping favorable outcomes. Remember, no overlapping means you’ve found disjoint events.

However, if a pair of events has even one favorable outcome, they are not disjoint. For example, if you are given a pair of events when rolling a number cube 1 – 6, multiple of 3 and odd. There is an overlapping favorable outcome. Since 3 is both a multiple of 3 and odd, these events are not disjoint.

When your child is given a pair of events, he/she can take notes to determine whether or not the events are disjoint. Visual learners may want to color-code or highlight. Here’s an example of how to identify a pair of events as disjoint or not.

Example 1: Suppose you draw one card from 10 cards that are numbered 1 – 10 and get a number that is greater than 5 and even.

Greater than 5: 6, 7, 8, 9, 10

Even: 2, 4, 6, 8, 10

These events are not disjoint. I know this because both lists show overlapping information for 6, 8, and 10, as highlighted.

Example 2: Suppose you draw one card from 10 cards that are numbered 1 – 10 and get a number that is greater than 5 and less than 2.

Greater than 5: 6, 7, 8, 9, 10

Less than 2: 1

These events are disjoint. I know this because there are no overlapping favorable outcomes (same possibilities).

(2)

In order to find the probability of a pair of disjoint, or mutually exclusive events, you will need to find the sum of the probabilities of each event. To find the sum, we will need to add the probabilities together. In order to do this, we will first need to determine the theoretical probability for each event. If you need to review how to find a theoretical probability, see Learning Tip 5. After finding each probability, we will add both of them together to find the probability of the disjoint events. Below is an example and step by step directions explaining how to find the probability of disjoint events.

Example: Suppose you toss a 1-6 number cube. What is the probability that you will toss a 5 or an even number.

1. Determine whether or not the events are disjoint, by making lists of all probabities.

5: 5

Even: 2, 4, 6

Since there are no overlapping favorable outcomes, move to step 2.

2. Write the theoretical probability for the first event. First, count all possible outcomes from the list on step one (above) and write your result as the numerator (top fraction). Next, determine the sample space and use it as the denominator (bottom fraction). Since the cube has 6 sides, our sample space is 6.

1/6

3. Use the steps above to write the theoretical probability for the next event.

3/6

4. Add up both probabilities.

1/6 + 3/6 = 4/6

5. Calculate your final answer as a percent. (optional)

4/6 67%

(3)

An Independent event is an event that does not affect the next event. Independent events are only possible if an experiment’s sample space is unchanged. In other words, you must always replace the item drawn or removed in the first experiment before conducting the second experiment. This will mean that probabilities from independent events will always have the same denominator, or sample space. If the item is not replaced, the sample space will change and the outcome will be affected.

Example 1: Tanya is drawing candies out of a bag of 12. She first gets a red, records her data and puts the candy back. On her second draw, she gets a green. This is an example of an independent event. Tanya replaced the candy before drawing a new one. This would give her the same denominator for each probability, 12. This is an independent event.

Non-example: Tanya is drawing candies out of a bag of 12. She first gets a red, records her data, and then eats it. Next, she draws a green. This is not an independent event, because her first denominator or sample space would be 12, but after she ate the red, the sample space would go down to 11 for the next event.

Be sure that your child understands that for an event to be independent, the sample space (denominator) must always be the same. This means the item must always be returned before the next event happens.

(4)

Independent probabilities can be calculated by finding the product of the probabilities of both events. In order to do this, you will need to be able to first, determine the probability of each independent event. Next, you will need to multiply the two probabilities together. Here’s a step-by-step example:

Example: Monica has a bag of marbles. Inside the bag are 2 red, 4 blue, 5 orange and 1 black marble. What is the probability that she will reach into the bag and get a red marble, then a blue marble? Find the P (r, b) or P (red or blue).

1. Count the total number of marbles in the bag. This will be your sample space. 12 is the sample space. This will be the denominator for your fraction because it give you the total possible outcomes.

2. Find the first probability, red. Look back at the original problem to determine how many marbles are red. Since 2 are red, there are two favorable outcomes for this event. This will be your numerator. Remember, the 12 for step one is your denominator.

2/12

3. Find the second probability. P (blue)

4/12

4. Multiply both probabilities together. P(red) x P(blue)

2/12 x 4/12 = 1/18

P (r, b) = 1/18 or about 5.5%

(5)

It is important that your child know important probability vocabulary terms for this skill. Be sure that your child understands and can explain what each of the terms below means.

Sample space – the total number of items in the experiment, also known as the number of all possible outcomes. The sample space is always the denominator or bottom number of the ratio (fraction) used to show a probabilty. Ex. If a baggie has a total of 12 marbles, the sample space is 12.

Favorable outcomes – the number of possible specified outcomes. This means, if you’re asked to find the probability for red, you asked to find the favorable outcomes for red, or more simply how many red there are in all. The favorable outcome is always the numerator for the ratio used to show a probability.

(6)

Children need to understand the theoretical probability of an event occurring. Probabilities range from 0 to 1, where 0 is impossible for an event to occur and 1 is 100% certain that an event will occur. Probabilities can be used to tell whether an event is; certain, very likely, equally likely, less likely and impossible to occur. Probabilities can be written as a fraction, a decimal or a percent. You may want to discuss how we hear probabilities in the media with weather forecasts, etc…

It is important that your child understand that when he/she finds theoretical probability, that he/she must always compare the number of favorable outcomes (probability being sought out) to the total number of possible outcomes (total possible combinations or choices).

The probability of an event is shown a P(n), where n is replaced with the event that is being solved for. So, if we were to be solving for the chances of spinning red on a spinner, we would write P(red) or P(r). Theoretical probability can be shown as a fraction. The numerator (top number) of the fraction will always be the number of favorable outcomes. The denominator (bottom number) of the fraction will always be the number of possible outcomes. The theoretical probability for any event would be shown as;

P(event) = number of favorable outcomes

number of all possible outcomes

The words would be replaced with numbers to solve for a specific probability. Here’s an example of how to use the formula above to find the theoretical probability of rolling an even number on a number cube 1-6.

Write the probability you are seeking in the parenthesis after the P.

P (even)

Count the possible favorable outcomes. The favorable outcomes for this problem are all even numbers on the cube. 2, 4, and 6 are even. Hence, there are 3 favorable outcomes of even numbers.

P(even) = 3__

Count all possible outcomes. There are 6 numbers on the cube, which means a total of 6 possible outcomes.

P(even) = 3__

6

If possible, write the fraction in simplest form by dividing both the numerator and denominator by the greatest common divisor (largest number that will divide into both evenly).

So, 3/6 is the theoretical probability of rolling an even number.

(7)

At times your child will be asked to write his/her probability as a percent. It is a good habit to be in to always do this. If your child needs a review on how to change a fraction into a percent, see below.

In order to turn a fraction into a percent, you will need to do two things.

Follow the steps to change the fraction into a decimal. (This step in simple terms is divide numerator by denominator). 3/12 = 3 12 = 0.25

Next, start at the decimal point and move it two places to the right. If the decimal stops behind the last number, you can just replace it with the percent sign. 0.25 = 25% However, if you have a decimal such as 0.375, you would keep the decimal and write 37.5%. Also, don’t forget, that if you have a decimal such as, 0.5 that you still need to move the decimal to places to the right. Since there is only one number to the right of the decimal, you add a 0 after the number. So, 0.5 is 50%.

Determine whether the events are disjoint or not. Explain how you know.

Suppose you draw one card from 10 cards that are numbered 1 – 10 and get a number that is a multiple of 5 and even.

(Not disjoint, 10 is both a multiple of 5 and even, so the favorable outcome of 10 overlaps.)

(2)

Determine whether the events are disjoint or not. Explain how you know.

Suppose you toss a cube with the numbers: 10, 15, 20, 25, 30, and 50 and get a number that is less than 15 and a multiple of 20.

(Disjoint, only 10 is less than 15 and 20 is the only multiple of 20. These don’t overlap.)

(3)

Determine whether the events are disjoint or not. Explain how you know.

You have picture cards of dogs and cats in a box. Each card has one animal on it. You pull one card from the box and you get a cat and a dog.

(Disjoint, you cannot pull a picture of a cat and a dog at the same time with one card, because there is only one animal on each card, either a cat or a dog.)

(4)

Determine whether the events are disjoint or not. Explain how you know.

Suppose a Bingo Machine is has tiles numbered 1 -100 inside and you draw out a multiple of 2 and a multiple of 5.

(Not disjoint, both 2 and five share multiples, such as: 10, 20, etc… These shared multiples are overlapping favorable outcomes.)

(5)

Determine whether the events are disjoint or not. Explain how you know.

Suppose a Bingo Machine is has tiles numbered 1 -100 inside and you draw out a prime number and an even number greater than 4.

(Disjoint, all prime numbers greater than 4 and less than 100 are odd, so there are no overlapping favorable outcomes.)

(6)

Calculate the probability of the disjoint events.

You roll a 1-6 number cube. What is the probability you will roll a composite number or a 1?

(4/6, about 67%)

(7)

Calculate the probability of the disjoint events.

You have a 16-section spinner numbered 1-16. What is the probability that you will spin an even number greater than 6 and a prime number?

(11/16, about 69%)

(8)

Calculate the probability of the disjoint events.

You have a 16-section spinner numbered 1-16. What is the probability that you will spin a number less than 6 and greater than 11?

(10/16, 62.5%)

(9)

Calculate the probability of the disjoint events.

You have letter tiles in a bag. The bag has the letters: M A T H I S F U N. What is the probability that you will pull out one tile that has an F or a vowel?

(4/9, about 44%)

(10)

Calculate the probability of the disjoint events.

Sadie put the names of her best friends on cards in a bag. The names are: Veronica, Steven, Tess, Brian, Mia, Mya, Dan, and Tony. What is the probability that she will draw one card with a name with 3 letters or that starts with a T?

(5/8, 62.5%)

(11)

Determine whether the events are independent or not. Explain how you know.

Joe is has a bag of colored candies. He pulls out one candy, records the data and puts the candy back in the bag. He then pulls out a second candy.

(Independent, he replaced the candy before doing the second event.)

(12)

Determine whether the events are independent or not. Explain how you know.

Amy is playing a card game with her sister. She draws a card, puts it in a new pile and then draws a second card.

(Not independent, Amy does not replace the card.)

(13)

Determine whether the events are independent or not. Explain how you know.

Talia captures of fish from her tank, records its color and puts it in a new tank. She then captures another fish.

(Not independent, Talia did not put the first fish back before catching the second fish.)

(14)

Determine whether the events are independent or not. Explain how you know.

Raheem pulled a card out of a hat, wrote down the color and returned it to the hat. He then shook the hat and pulled a second card.

(Independent, the card was replaced between events.)

(15)

Determine whether the events are independent or not. Explain how you know.

Mark spun and spinner and rolled a number cube at the same time.

(Independent, one event did not affect the other.)

(16)

Calculate the probability of the independent events.

You have 5 orange, 5 red, 12 blue, and 3 green popsicles in a bag. What is the probability that your first friend will pull out a red without looking and that your second friend will pull out a green?

(3/125, 2.4%)

(17)

Calculate the probability of the independent events.

You have 5 orange, 5 red, 12 blue, and 3 green popsicles in a bag. What is the probability that your first friend will pull out a blue without looking and that your second friend will pull out a blue?

(124/625, 19.84%)

(18)

Calculate the probability of the independent events.

You have 5 orange, 5 red, 12 blue, and 3 green popsicles in a bag. Find P (g, r, o)

(3/625, 0.48%)

(19)

Calculate the probability of the independent events.

You have a spinner with four equal sections: red, yellow, green, blue. You also have a 1-6 number cube. Find P (y, even).

(1/8, 12.5%)

(20)

Calculate the probability of the independent events.

You have a spinner with four equal sections: red, yellow, green, blue. You also have a 1-6 number cube. Find P (r, 2).

(1/24, about 4%)

(21)

You have a spinner with four equal sections: red, yellow, green, blue. You also have a 1-6 number cube. Find P (g or b, odd).

(1/4, 25%)

(22)

You have a spinner with four equal sections: red, yellow, green, blue. You also have a 1-6 number cube. Find P (g, y or b, 6).

(1/8, 12.5%)

(23)

You flip a coin and toss a 1-6 number cube at the same time. Find

P(heads, even)

(1/4, 25%)

(24)

A carnival game has 10 red cups, 25 yellow cups and 15 purple cups. The prizes awarded for red cups are stuffed animals, yellow cups, no prize and purple cups, a poster. What are the chances that you will get no prize on your first try and a stuffed animal on your second try?