Question

Marcos is doing a card trick using a standard $ 52 $ card deck. He shows his friend a card, replaces it, and then shows his friend another card. What is the probability that the first card is club and the second card is not a heart?

Answer 1

Hint: In this question we have to calculate the probability of the given conditions happening the given conditions are
I.That first card is club
II.That the second card is not a heart card.
For this we know that these are mutually exclusive i.e. they are independent events and therefore there probability can be individually calculated and since both the given conditions must be true we have to multiply the calculated probability instead of adding them (we add probabilities when the or conditions is given ie only one of the events should happen). The probability of both A) and B) can be calculated using the standard formula for calculating probability.

Complete step-by-step answer:
Since there are $ 13 $ clubs in a deck, the probability of finding a club in $ 52 $ cards is
 \[ = \dfrac{{13}}{{52}}\] , which can be written as
 \[ = \dfrac{1}{4}\]
The same goes for the probability of not finding a heart, there are $ 39 $ cards in a deck which are not hearts so the probability of such arrangement is
 \[ = \dfrac{{39}}{{52}}\]
Which will be written as
 \[ = \dfrac{3}{4}\]
Thus we got both our probabilities but since we have to find the condition of both happening and these are independent events we can write:
 \[ = \dfrac{1}{4} \times \dfrac{3}{4}\]
 \[ = \dfrac{3}{{16}}\]
The final answer is \[\dfrac{3}{{16}}\]

Note: For mutually exclusive events we calculate probabilities independently not as conditionally. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.