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
Well, well, well, let's dive into the world of cards, shall we? P(A|(not B)) is the probability of selecting a diamond given that a club is not selected.
To figure this out, we need to find two probabilities: P(A) and P(not B).
P(A) is the probability of selecting a diamond from a deck of 52 cards. There are 13 diamonds in total (one for each value), so the probability is 13/52, which simplifies to 1/4.
Now, let's calculate P(not B). There are 13 clubs in the deck, so the probability of not selecting a club is 39/52, which simplifies to 3/4.
To find P(A|(not B)), we need to multiply P(A) by P(not B):
P(A|(not B)) = P(A) * P(not B) = (1/4) * (3/4) = 3/16.
So there you have it! The probability of selecting a diamond given that a club is not selected is 3/16.
To determine the probability of event A (a diamond is selected) given event B (a club is not selected), we need to use Bayes' theorem.
Bayes' theorem states:
P(A | B) = (P(B | A) * P(A)) / P(B)
In this case, we want to find P(A | (not B)), which means we want to find the probability of event A given that event B did not occur.
To find this probability, we need to calculate:
P(A | (not B)) = (P((not B) | A) * P(A)) / P(not B)
Now let's break down each probability component:
P((not B) | A): Probability that a club is not selected given that a diamond is selected.
Since each card selection is independent, once we have chosen a diamond (13 out of 52 cards), that leaves us with 39 non-club cards out of the remaining 51 cards. Therefore, P((not B) | A) = 39/51.
P(A): Probability of selecting a diamond.
There are 4 diamonds in a deck of 52 cards. Therefore, P(A) = 4/52.
P(not B): Probability of not selecting a club.
There are 13 clubs in a deck of 52 cards. Therefore, P(not B) = 39/52.
Putting it all together, we can calculate the final probability:
P(A | (not B)) = (P((not B) | A) * P(A)) / P(not B)
= (39/51 * 4/52) / (39/52)
= 4/51
Therefore, P(A | (not B)) is equal to 4/51.
To determine P(A|(not B)), we need to find the probability that a diamond is selected given that a club is not selected.
We can start by calculating the individual probabilities:
P(A) = probability of selecting a diamond = 13/52 = 1/4 (as there are 13 diamonds in a deck of 52 cards)
P(B) = probability of selecting a club = 13/52 = 1/4 (as there are 13 clubs in a deck of 52 cards)
Now, to find P(A|(not B)), we need to find the probability of selecting a diamond given that a club is not selected.
Let's calculate this step by step:
1. Determine the probability that a club is not selected:
P(not B) = 1 - P(B) = 1 - 1/4 = 3/4
2. Determine the probability of selecting a diamond given that a club is not selected:
P(A|(not B)) = P(A and (not B)) / P(not B)
To find P(A and (not B)), we need to calculate the probability of both events happening simultaneously.
Since selecting a diamond and not selecting a club are independent events, the probability of both happening is the product of their individual probabilities:
P(A and (not B)) = P(A) * P(not B) = (1/4) * (3/4) = 3/16
Now we can substitute these values into the formula:
P(A|(not B)) = P(A and (not B)) / P(not B) = (3/16) / (3/4)
To divide fractions, we can multiply by the reciprocal:
P(A|(not B)) = (3/16) * (4/3) = 12/48 = 1/4
Therefore, P(A|(not B)) = 1/4.
In conclusion, the probability of selecting a diamond given that a club is not selected is 1/4.