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Mutually Exclusive Events

Probability of two events that cannot occur at the same time P(A or B) = P(A) + P(B)

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Union of Compound Events

If you think it through, it should make sense that the probability of pulling one Queen at random from a standard deck is \begin{align*}\frac{4}{52}\end{align*}452 or \begin{align*}\frac{1}{13}\end{align*}113, since there are 4 Queens in a standard 52 card deck. How then would you calculate the probability of pulling a Queen OR a King from the same deck?

Credit: Jamie
Source: https://www.flickr.com/photos/jamiesrabbits/5791400156/
License: CC BY-NC 3.0

Union of Compound Events 

When multiple independent events may occur during a particular experiment, there are a couple of different types of outcomes you may need to consider:

  • Intersection: the probability of both or all of the events you are calculating happening at the same time (less likely).
  • Union: the probability of any one of multiple events happening at a given time (more likely).

In this lesson, we will focus on union. Calculating the union is relatively easy, you just add up the individual probabilities of the events:

\begin{align*}P(x \ or \ y)=P(x)+P(y)\end{align*}P(x or y)=P(x)+P(y)

This can also be thought of as:

\begin{align*}P(x \ or \ y)=\frac{(\text{number of outcomes where} \ x \ \text{is true}) +(\text{number of outcomes where} \ y \ \text{is true})}{\text{total number of possible outcomes}}\end{align*}P(x or y)=(number of outcomes where x is true)+(number of outcomes where y is true)total number of possible outcomes

It is really just that simple! It is intuitive also, assuming there is no overlap (which we will consider later), it just makes sense to think that if you have a 20% probability of one thing happening, and a 30% probability of another, then you have a 50% probability of one of the two of them happening during a given experiment.

 

Calculating Probability 

1. You are given a big containing 15 equally sized marbles. You know there are 5 yellow marbles, 5 blue marbles, and 5 green marbles in the bag. What is the statistical probability that you would pull a yellow or green marble out, if you reach in the bag and grab a marble at random?

Recall the formula for the union of simple probabilities:

\begin{align*}P(x \ or \ y)=\frac{(\text{number of outcomes where} \ x \ \text{is true}) +(\text{number of outcomes where} \ y \ \text{is true})}{\text{total number of possible outcomes}}\end{align*}P(x or y)=(number of outcomes where x is true)+(number of outcomes where y is true)total number of possible outcomes

In this case, we have:

\begin{align*}P(yellow \ or \ green)=\frac{5 \ yellow \ marbles+5 \ green \ marbles}{15 \ total \ marbles}\end{align*}P(yellow or green)=5 yellow marbles+5 green marbles15 total marbles

Which would reduce to:

\begin{align*}P(yellow \ or \ green)=\frac{2}{3} \ or \ 66.6 \bar{6} \%\end{align*}P(yellow or green)=23 or 66.66¯%

2. What is the probability of rolling an odd or even number on a standard six-sided die?

A standard die has three odd numbers (1, 3, 5) and three even numbers (2, 4, 6). Therefore, the probability of rolling an odd or even number is:

\begin{align*}P(odd \ or \ even)=\frac{3 \ odd + 3 \ even}{6 \ total}=\frac{6}{6}\end{align*}P(odd or even)=3 odd+3 even6 total=66

Reducing to:

\begin{align*}P(odd \ or \ even)=1 \ or \ 100 \%\end{align*}P(odd or even)=1 or 100%

3. If Lawrence is playing with a standard 52-card deck, what is the probability of pulling a 2, a 4, or a 6 out of the deck at random?

 Let’s solve this one as the total of the individual probabilities. Lawrence’s probability of pulling a 2, 4, or 6 is the same as the union of the probability of each possible outcome:

\begin{align*}P(2, 4, \ or \ 6)=P(2)+P(4)+P(6)=\frac{1}{13}+\frac{1}{13}+\frac{1}{13}=\frac{3}{13} \ or \ 23.1 \%\end{align*}P(2,4, or 6)=P(2)+P(4)+P(6)=113+113+113=313 or 23.1%

Earlier Problem Revisited

It should make sense now that the probability of pulling one Queen at random from a standard deck is \begin{align*}\frac{4}{52}\end{align*}452 or \begin{align*}\frac{1}{13}\end{align*}113, since there are 4 Queens in a standard 52 card deck. How then would you calculate the probability of pulling a Queen OR a King from the same deck?

Remember that the union of multiple probabilities is simple the total sum of all of the individual probabilities:

\begin{align*}P(Queen \ or \ King)=\frac{4 \ Queens+4 \ Kings}{52 \ Cards}=\frac{8}{52} \ or \ \frac{4}{26} \ or \ \frac{2}{13}=15.4 \%\end{align*}P(Queen or King)=4 Queens+4 Kings52 Cards=852 or 426 or 213=15.4%

 

Examples 

Example 1

What is the statistical probability of pulling either the only red or the only blue marble out of a bag with 12 marbles in it?

\begin{align*}P(red \ or \ blue)=\frac{1\text{ red marble} + 1\text{ blue marble}}{12\text{ total marbles}}=\frac{2}{12} \ or \ \frac{1}{6} \ or \ 16.6 \%\end{align*}P(red or blue)=1 red marble+1 blue marble12 total marbles=212 or 16 or 16.6%

Example 2

What is the probability of a spinner landing on “2”, “3”, or “6” if there are 6 equally spaced points on the spinner?

\begin{align*}P(2 \ or \ 3 \ or \ 6)=\frac{1 \text{ number }6 + 1\text{ number }2 + 1\text{ number }3}{6\text{ total numbers}}=\frac{3}{6} \ or \ \frac{1}{2} \ or \ 50 \%\end{align*}P(2 or 3 or 6)=1 number 6+1 number 2+1 number 36 total numbers=36 or 12 or 50%

Example 3

What is the probability of pulling a red or black card at random from a standard deck?

\begin{align*}P(red \ or \ black)=\frac{26\text{ red cards} + 26\text{ black cards}}{52\text{ total cards}}=\frac{52}{52}=\frac{1}{1} \ or \ 100 \%\end{align*}P(red or black)=26 red cards+26 black cards52 total cards=5252=11 or 100%

Example 4

What probability of picking a red or green marble from a bag with 5 red, 7 green, 6 blue, and 14 yellow marbles in it?

\begin{align*}P(red \ or \ green)=\frac{5\text{ red marbles}+7\text{ green marbles}}{32\text{ total marbles}}=\frac{12}{32} \ or \ \frac{6}{16} \ or \ \frac{3}{8} \ or \ 37.5 \%\end{align*}P(red or green)=5 red marbles+7 green marbles32 total marbles=1232 or 616 or 38 or 37.5%

Example 5

What is the of shaking the hand of a student wearing red if you randomly shake the hand of one person in a room containing the following mix of students?

  • 13 female students wearing blue
  • 7 male students wearing blue
  • 6 female students wearing red
  • 9 males students wearing red
  • 18 female students wearing green
  • 21 male students wearing green

\begin{align*}P(red)=\frac{6\text{ females wearing red}+9\text{ males wearing red}}{74\text{ total students}}=\frac{15}{74} \ or \ 20.3 \%\end{align*}P(red)=6 females wearing red+9 males wearing red74 total students=1574 or 20.3%

Review 

  1. What is the probability of rolling a standard die and getting between a 1 and 6 (inclusive)?
  2. What is the probability of pulling one card from a standard deck and it being an 8, a 3, or a queen?
  3. What is the probability of rolling a 5 or a 2 on an 8-sided die?
  4. What is the probability of pulling one card from a standard deck and it being a spade, a diamond, or a club?
  5. What is the probability of rolling a 1, 3, or 5 on a 7-sided die?
  6. What is the probability of pulling one card from a standard deck and it being a king, a 4, or a 8?
  7. What is the probability of pulling a yellow or blue candy from a bag containing 35 candies equally distributed among yellow, blue, green, red, and brown candies?
  8. What is the probability of spinning 2, 4, or 7 on a 10-space spinner (equally spaced)?
  9. What is the probability of rolling a 1, 3, 5, or 6 on a 20-sided die?
  10. A car factory creates cars in the following ratio: 3 green, 2 blue, 7 white, 2 black and 1 brown. What is the probability that a randomly selected car will be either blue or brown?
  11. There are 4 flavors of donuts on the shelf: glazed, sprinkles, plain, and powdered sugar. If there are equal numbers of each of the non-plain donuts, and half as many plain as any one of the others, what is the probability of randomly choosing a plain donut out of all donuts on the shelf?
  12. What is the probability of randomly choosing a red Ace or a black King from a standard deck?
  13. What is the probability of rolling a prime number or an even number on a standard die?
  14. Mr. Spence’s class has 13 students. 4 students are wearing coats, 3 are wearing vests, 3 are wearing hoodies, and the rest are in t-shirts. What is the probability that Mr. Spence will randomly call the name of a student wearing a coat or a vest?
  15. In the same class, what is the probability that Mr. Spence will randomly call the name of a student in a hoodie or t-shirt?

Review (Answers)

To view the Review answers, open this PDF file and look for section 6.2. 

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Vocabulary

Event

An event is a set of one or more possible results of a probability experiment.

Independent Events

Two events are independent if the occurrence of one event does not impact the probability of the other event.

Intersection

Intersection is the probability of both or all of the events you are calculating happening at the same time (less likely).

Mutually Exclusive Events

Mutually exclusive events have no common outcomes.

Outcome

An outcome of a probability experiment is one possible end result.

Permutation

A permutation is an arrangement of objects where order is important.

statistical probability

A statistical probability is the calculated probability that a given outcome will occur if the same experiment were completed an infinite number of times.

union

\cup is a symbol that stands for union and is used to connect two groups together. It is associated with the logical term OR.

Image Attributions

  1. [1]^ Credit: Jamie; Source: https://www.flickr.com/photos/jamiesrabbits/5791400156/; License: CC BY-NC 3.0

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