Question

Suppose we draw eight cards from a pack of 52 cards. What is the probability of getting less than three spades?​

Answer 1

Answer:

Suppose we draw eight cards from a pack of 52 cards. What is the probability of getting less than three spades? ... N = 52 since there are 52 cards in a deck. k = 13 since there are 13 spades in a deck.

Answer 2

Answer:

The probability of getting less than 3 spades is 0.685627.

Step-by-step explanation:

Given:

Drawing  8 cards from a pack of 52 cards.

Getting less than three spades means that we get 0 cards, 1 card, or  2 cards.

In a deck of 52 cards, 13 cards are spades, and 52 - 13 = 39 are nonspade cards.

While drawing 8 cards from the pack one possibility is getting no spade at all and all 8 are non-spades. 2nd  possibility is getting 1 spade and 7 non-spades. the third possibility is that we get 2 spades and 6 non-spades.

The binomial coefficient   $\left(\begin{array}{c}n \\k\end{array}\right)$  is read as "n choose k."

Note that there is 1 way to choose no cards from 13 and there are 13 ways to choose 1 card from 13, and so,

                                  $\left(\begin{array}{c}13 \\0\end{array}\right)=1 \text { and }\left(\begin{array}{c}13 \\1\end{array}\right)=13 .$

Then

$\mathrm{P} (less than 3 spades drawing 8 cards) =$

$\frac{\left(\begin{array}{c}13 \\ 0\end{array}\right)\left(\begin{array}{c}39 \\ 8\end{array}\right)}{\left(\begin{array}{c}52 \\ 8\end{array}\right)}+\frac{\left(\begin{array}{c}13 \\ 1\end{array}\right)\left(\begin{array}{c}39 \\ 7\end{array}\right)}{\left(\begin{array}{c}52 \\ 8\end{array}\right)}+\frac{\left(\begin{array}{c}13 \\ 2\end{array}\right)\left(\begin{array}{c}39 \\ 6\end{array}\right)}{\left(\begin{array}{c}52 \\ 8\end{array}\right)}$

$=\frac{\left(\begin{array}{c}39 \\8\end{array}\right)+13\left(\begin{array}{c}39 \\7\end{array}\right)+\left(\begin{array}{c}13 \\2\end{array}\right)\left(\begin{array}{c}39 \\6\end{array}\right)}{\left(\begin{array}{c}52 \\8\end{array}\right)} =\frac{259407}{378350} \approx .685627$

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