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A 5-card hand is dealt from a deck of 52 cards. What is the probability that all are hearts?

Solution:

Probability defines the likelihood of occurrence of an event.

There are many real-life situations in which we may have to predict the outcome of an event.

We may be sure or not sure of the results of an event.

In such cases, we say that there is a probability of this event to occur or not occur.

It is given that,

A 5-card hand is dealt from a deck of 52 cards = ⁵²C₅ ways

So,

n = ⁵²C₅

Let us assume E be the event of getting 5 cards are hearts,

There are total 13 cards of hearts in a deck of 52 cards,

Therefore,

n(E) = ¹³C₅

As we know that probability is defined as total number of favourable outcomes divided by total number of possible outcomes.

Hence,

P(E) = ¹³C₅ / ⁵²C₅

Therefore, the probability that all are hearts is ¹³C₅ / ⁵²C_5.

Summary:

A 5-card hand is dealt from a deck of 52 cards. The probability that all are hearts is ¹³C₅ /⁵²C₅.

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