Compound Event
Definition: Involving two or more separate events
Independent
Definition: The occurrence of one event has no effect on the occurrence of another,
Product rule for independent event
A coin is flipped while a die is rolled. What is the probability of flipping heads and rolling 5 in a single trial?
Method: Flipping head = 1/2
Rolling 5 in a single trial = 1/6
Answer: P(A and B) = P(A) x P(B) = 1/2 x 1/6
Method: Flipping head = 1/2
Rolling 5 in a single trial = 1/6
Answer: P(A and B) = P(A) x P(B) = 1/2 x 1/6
Dependent
Definition: The conditional probability of B, P(B|A), is the probability that B occurs, given that A has already occurred.
Product rule for dependent event
A professional hockey team has eight wingers. Three of these wingers are 30-goal scorers, or "snipers". Every fall the team plays an exhibition match with the club's farm team. In order to make the match more interesting for the fans, the coaches agree to select two wingers at random from the pro team to play for the farm team. What is the probability that two snipers will play for the farm team?
Method: Probability that two snipers will play for the farm team = 3/8 = P(A)
Probability that two sniper will play for the farm team if one sniper is already selected = 2/7
= P(B|A)
Answer: P(A and B) = P(A) x P(B|A) = 3/8 x 2/7
Method: Probability that two snipers will play for the farm team = 3/8 = P(A)
Probability that two sniper will play for the farm team if one sniper is already selected = 2/7
= P(B|A)
Answer: P(A and B) = P(A) x P(B|A) = 3/8 x 2/7
Mutually Exclusive
Definition: Events cannot occur at the same time.
Addition rule for mutually exclusive events
Teri attends a fundraiser at which 15 T-shirts are being given away as door prizes. Door prize winner are randomly given a shirt from a stock of 2 black shirts, 4 blue shirts, and 9 white shirts. Teri really likes the black and blue shirts, but is not too keen on the white ones. Assume that Teri wins the first door prize, what is the probability that she will get a shirt that she likes?
Think: She can get only one shirt! She cannot take two shirts. It is not a dependent event. It is an independent event because the probability of getting blue does not change probability of getting black. However, it cannot happen at the same time.
Method: Probability of having black shirt = 2/15 = P(A)
Probability of having blue shirt = 4/15 = P(B)
Answer: P(A or B) = P(A) + P(B) = 2/15 + 4/15
Think: She can get only one shirt! She cannot take two shirts. It is not a dependent event. It is an independent event because the probability of getting blue does not change probability of getting black. However, it cannot happen at the same time.
Method: Probability of having black shirt = 2/15 = P(A)
Probability of having blue shirt = 4/15 = P(B)
Answer: P(A or B) = P(A) + P(B) = 2/15 + 4/15
Addition rule for non-mutually exclusive events
A card is randomly selected from a standard deck of cards. What is the probability that either a heart or a face card (jack, queen, or king) is selected?
Think: If you receive a face card (jack, queen or king) of hearts, you have a heart and a face card. So, it is not a mutually exclusive event. The main question is how can we decide the method we should use to solve this question. Is it independent? yes! It is independent. Is there another distinct event other than getting a card? NO! Then you use Addition Rule
Method: P(A) is having a heart = 13/52
P(B) is having a face card = 12/52
P(A and B) is having a face card of hearts = 3/52
Then P(A or B) = P(A) + P(B) - P(A and B) = 13/52 + 12/52 - 3/52
Think: If you receive a face card (jack, queen or king) of hearts, you have a heart and a face card. So, it is not a mutually exclusive event. The main question is how can we decide the method we should use to solve this question. Is it independent? yes! It is independent. Is there another distinct event other than getting a card? NO! Then you use Addition Rule
Method: P(A) is having a heart = 13/52
P(B) is having a face card = 12/52
P(A and B) is having a face card of hearts = 3/52
Then P(A or B) = P(A) + P(B) - P(A and B) = 13/52 + 12/52 - 3/52