Dependent events - events whose outcomes are affected by each other.
Example: Drawing an item, then drawing another item,without replacing the first item, results in a pair of dependent events.
Conditional Probability - the probability of an event occurring given that another event has already occurred.
A group of men and women are surveyed to determine their choice of pop. The results are as follows:
1. What is the probability a randomly selected person prefers coke?
Is this conditional or non-conditional probability? ___________________________
2. What is the probability a randomly selected person prefers Coke, given that they are male?
Is this conditional or non-conditional probability? ___________________________
3. What is the probability a randomly selected person is male, given that they prefer Coke?
Is this conditional or non-conditional probability? ___________________________
4. What is the probability a randomly selected person prefers Pepsi, given that they are female?
Is this conditional or non-conditional probability? ___________________________
The notation for conditional probability is and is read, "the probability that event B will occur, given that event A has already occurred." The formula is:
Prefers Pepsi Prefers Coke Total
Male 27 5 32
Female 30 38 68
Let's apply this formula to the last question:
What is the probability a randomly selected person prefers Pepsi, given that they are female?
Sometimes, we may be given the conditional probability and we want to find the probability of both events occurring (the intersection of the two events).
To do this, we can re-arrange the formula for conditional probability to solve for :
If event B depends on event A occurring, then the probability that both events will occur can be represented as follows:
Ex (1) Amy draws a card from a well-shuffled deck of 52 playing cards. Then she draws another card from the deck without replacing the first card.
a) Are these two events dependent or independent?_____________________ b) Determine the probability that both cards are face cards.
Ex (2) Amy draws a card from a well-shuffled deck of 52 playing cards. Then she puts the card back in the deck, shuffles again, and draws another card from the deck.
a) Are these two events dependent or independent? _____________________ b) Determine the probability that both cards are face cards.
Ex (3) Jordan has a bag that contains 4 nickels and 7 quarters. He pulls out one coin at random and then another coin, without replacing the first coin. Determine the probability of each event below.
a) He pulls out a pair of nickels. b) He pulls out a pair of quarters.
Ex (4) A computer manufacturer knows that in a box of 80 computer chips, 4 will be defective. Daniel draws 2 chips, at random, from a box of 80 chips. Determine the probability that Daniel will draw the following:
a) 2 defective chips b) 2 non-defective chips
c) Exactly 1 defective chip.
Ex (5) There are 112 males and 133 females in this year's graduating class. Of these students, 24 males and 32 females plan to attend UPEI next year.
a) Determine the probability that a randomly selected student plans to attend UPEI.
b) A randomly selected student plans to attend UPEI. Determine the probability that the selected student is male.
c) A female student is randomly selected. Determine the probability that the selected student is going to UPEI.
Ex (6) At an office, there are 100 male and 120 female employees. Twenty of the males and 30 of the females have no siblings. An employee is randomly selected.
b) An employee is chosen that has no siblings. Find the probability that the employee is female.
Ex (7) At the Humane Society, there are 42 cats and 18 dogs. Of these animals, 20 cats and 7 dogs had been captured and taken to the shelter as strays. And animal is randomly chosen.
a) The chosen animal was a stray. Find the probability that the animal is a dog.
b) Find the probability that the animal was not a stray given that it is a cat.
Ex (1) Anna is the coach of a rugby team. Based on the team's record, it has a 65% chance of winning in the rain and a 75% chance of winning on nice days. Tomorrow, there is a 20% chance of rain. There are no ties in this rugby league. What is the probability that Anna's team will win tomorrow?
Ex (2) You are on a soccer team, and you prefer to be the goalkeeper. However, your coach decides what position you will play, and you have two coaches. With coach Sam, your probability of being the goalkeeper is 40%. With coach Alex, this probability is 30%.
Recall the formula used to calculate Conditional Probability:
Ex (3) Brett travels to work by car everyday. He carpools 80% of the time and travels alone 20% of the time. When he travels in a carpool, he is early for work 10% of the time. When he travels alone, he is early 25% of the time. Brett was early for work
today. What is the probability that he travelled to work alone?
Ex (4) Megan has math homework every night. She does her homework 70% of the time. When she does her homework, she understands the lesson the next day 90% of the time. When she doesn't do her homework, she understands the lesson 60% of the time.
