Question

You have a standard deck of 52 playing cards. If you draw two cards at random from the deck.

Required:
a. What is the probability that they are both hearts?
b. Your cat ate one of the playing cards! Assuming that the eaten card was a diamond, what is the probability that two cards drawn randomly from the deck are both hearts?
c. Are the events two cards drawn at random are both heartsl and the missing card is a diamond) independent? Fully justify your answer with math.

Answer 1

Explained below.

Explanation:

In mathematics, the procedure to select k items from n distinct items, without replacement, is known as combinations.

The formula to compute the combinations of k items from n is given by the formula:

`{n\choose k}=(n!)/(k!\cdot (n-k)!)`

(a)

Compute the probability that the two cards selected are both hearts as follows:

`P(2\ Hearts)=\frac{{13\choose 2}}{{52\choose 2}}`

`=(78)/(1326)\\\\=0.05882353\\\\\approx 0.0588`

Thus, the probability that the two cards selected are both hearts is 0.0588.

(b)

It is provided that one of the card was eaten and it was a diamond.

So, now there are 51 cards in the deck.

And of these 51 cards two are selected.

Compute the probability that the two cards selected are both hearts as follows:

`P(2\ Hearts)=\frac{{13\choose 2}}{{51\choose 2}}`

`=(78)/(1275)\\\\=0.06117647\\\\\approx 0.0612`

Thus, the probability that the two cards selected are both hearts is 0.0612.

(c)

No, the events two cards drawn at random are both hearts and the missing card is a diamond are not independent.

This is because the probability of selecting two hearts is dependent on the total number of cards available.