Question

One way to interpret conditional probability is that the sample space of the conditional probability becomes the "conditioning" event. If Event A is drawing a 10 from a deck of cards and Event B is drawing a "spade" from a deck of cards, what would be the sample space for the conditional probability P(A|B)?

Answer 1

Answer:

The sample space is the set of elements in B i.e n(B) = 13

Explanation:

The total number of cards in a deck of cards is 52

The number of cards that carry the number 10 is 4

The number of space cards is 13

The probability of the first event is thus 4/52 = 1/13

The probability of the second event is 13/52 = 1/4

Now, there is only a space card that carries the number 10

This means that the intersection of both events contains a single element in its sample space: n(A n B) = 1

We have the conditional probability as:

$P(A|B)\text{ = }\frac{P(A\text{ }\cap\text{ B\rparen}}{P(B)}$

Substituting the values, we have it that:

$P(A|B)\text{ = }\frac{\frac{1}{52}}{\frac{1}{4}}\text{ = }\frac{1}{52}\times4\text{ = }\frac{1}{13}$

The denominator value 13 is the sample space for the conditional probability and that is the n(B) sample space3