A card is drawn from a standard deck of cards. Find P(A U B) in each part.
A= {getting a heart}, B= {getting an even number}
A= {getting a club}, B= {getting a red card}
A= {getting an ace}, B= getting a black card}
A= {getting a prime}, B= {getting a diamond}
Can you please show me how to work this
In all the cases, the probabilities of P(A) are not mutually exclusive from P(B). For example, we can get a heart AND an even number, which is P(A∩B).
The event of getting a heart OR an even number is therefore obtained by the addition rule:
P(A∪B)=P(A)+P(B)-P(A∩B)
P(A)=13 hearts out of 52=13/52=1/4
P(B)=6(assuming the queen is 12) /13 (for each suit)=6/13
P(A∩B)=#even hearts/52=6/52=3/26
So
P(A∪B)=P(A)+P(B)-P(A∩B)
=1/4+6/13-3/26
=(14+24-6)/52
=8/13
The other ones are all similar.
To find the probability of the union of two events (P(A U B)), you need to calculate the probability of Event A occurring, the probability of Event B occurring, and subtract the probability of both events occurring at the same time. The formula for P(A U B) is given by:
P(A U B) = P(A) + P(B) - P(A ∩ B)
where P(A) is the probability of Event A occurring, P(B) is the probability of Event B occurring, and P(A ∩ B) is the probability of both events A and B occurring simultaneously.
Let's calculate P(A U B) for each given scenario.
1. A = {getting a heart}, B = {getting an even number}:
- P(A) = Number of hearts in a deck / Total number of cards in a deck = 13/52 = 1/4 (since there are 13 hearts in a deck of 52 cards)
- P(B) = Number of even-numbered cards in a deck / Total number of cards in a deck = 20/52 = 5/13 (since there are 10 even-numbered cards in each suit, and there are 4 suits)
- P(A ∩ B) = Number of hearts that are even-numbered / Total number of cards in a deck = 5/52
Now we can substitute these values into the formula:
P(A U B) = P(A) + P(B) - P(A ∩ B)
P(A U B) = (1/4) + (5/13) - (5/52)
P(A U B) = 13/52 + 20/52 - 5/52
P(A U B) = 28/52
P(A U B) = 7/13
So, the probability of drawing a heart or an even-numbered card from a standard deck is 7/13.
You can follow the same steps to calculate P(A U B) for the other scenarios provided. Remember to determine the probabilities of A, B, and A ∩ B in each case.