Selecting Cards Three cards are randomly selected from a standard deck of 52 -card deck, one at a time with each card replaced in the deck before the next one is picked. Find the probability of each event

(a) All three cards are hearts.

(b) Exactly two of the cards are spades.

(c) None of the cards is a diamond.

(d) At least one of the cards is a club.

(a)

To find:

The probability that all three cards are hearts.

Given:

Three cards are randomly selected from a standard deck of 52-card deck, one at a time with each card replaced in the deck before the next one is picked. Find the probability of each event

Approach:

The formula for the probability of success is,

p=Favourable OutcomesTotal Outcomes …(1)

The formula for getting exactly r successes in n independent trials is,

P(X=r)=C(n,r)pr(1−p)n−r …(2)

Here, X is a random variable.

The formula for the complement of event E is,

P(E′)=1−P(E) …(3)

Here, E′ denotes the complement of E.

Calculation:

Let X be the event that the card is a heart

(b)

To find:

The probability that exactly two of the cards are spades.

(c)

To find:

The probability that none of the card is a diamond.

(d)

To find:

The probability that at least one of the card is a club.