Suppose one card is selected at random from an ordinary deck of 52 playing cards.
Let
A= event a diamond is selected
B=event a club is selected
Determine P(A|(not B))
Could some one explain this to me and tell me how they got the answer
if NOt B, then the card has to be a diamond, heart, or spade.
Pr(A|NOT B)=1/3
To determine the probability of event A (a diamond is selected) given that event B (a club is selected) does not occur, we need to use conditional probability.
First, let's find the probability of event (not B), which represents the event that a club is not selected.
There are 52 cards in a deck, and 13 of them are clubs. Therefore, the probability of not selecting a club is:
P(not B) = 1 - P(B) = 1 - (number of clubs / total number of cards) = 1 - (13/52) = 3/4
Next, we can calculate the probability of event A given that (not B) occurred, denoted as P(A|(not B)).
To find this probability, we need to consider the sub-sample space that excludes clubs, which contains 39 cards (52 total cards - 13 clubs). Among those 39 cards, there are 13 diamonds (since there are 13 cards of each suit in a deck). Therefore, the probability of event A given that (not B) occurred is:
P(A|(not B)) = (number of favorable outcomes) / (number of possible outcomes)
= 13/39
= 1/3
Hence, the probability of event A occurring given that event B does not occur is 1/3 or approximately 0.3333.
To determine the probability of event A (selecting a diamond) given that event B (selecting a club) does not occur, we can use the conditional probability formula:
P(A|(not B)) = P(A and (not B)) / P(not B)
First, let's find P(A and (not B)), which represents the probability of selecting a diamond but not a club from the deck.
In a standard deck of 52 playing cards, there are 13 diamonds. Since there are 4 suits in total and each suit has 13 cards, the probability of selecting a diamond is:
P(A) = 13 / 52
Next, we need to find the probability of not selecting a club (not B). There are 13 clubs in the deck, so the probability of selecting a club is:
P(B) = 13 / 52
However, we want the probability of not selecting a club, which is simply the complement of P(B):
P(not B) = 1 - P(B) = 1 - (13 / 52) = 39 / 52
Finally, we can calculate P(A|(not B)) using the conditional probability formula:
P(A|(not B)) = P(A and (not B)) / P(not B) = (13 / 52) / (39 / 52) = 13 / 39 = 1/3
Therefore, the probability of selecting a diamond given that a club is not selected is 1/3.