Question

A bridge hand is defined as 13 cards selected at random and without replacement from a deck of 52 cards. In a standard deck of cards, there are 13 cards from each suit: hearts, spades, clubs, and diamonds. What is the probability of being dealt a hand that does not contain a heart?

a. What is the group of interest?

b. How many are in the group of interest?

c. How many are in the other group?

d. Let X = _________. What values does X take on?

e. The probability question is P(_______).

f. Find the probability in question.

g. Find the (i) mean and (ii) standard deviation of X

Short answer

a. The group of interest is the pack of heart cards.

b. Cards that are not hearts is$39$

c. Other group includes the number of hearts in a deck which is $13$

d. X is the number of cards that are not hearts and the value of X is $0,1,2,3,......,13.$

e. The probability is$P(X=13)$

f. The probability is $0.0128$

g. mean is $9.75$and standard deviation is $1.3653$

Step-by-step solution

Content Introduction

A discrete probability distribution, the hypergeometric distribution. It's utilised when you want to know how likely it is to get a certain number of successes from a given sample size without replacement.

Explanation (part a)

The group of interest is the pack of heart cards.

Explanation (part b)

The group of interest is hearts.

The pack of cards contain$52$in total and the pack contains the deck of hearts is$13$, therefore, the total card$52-13=39$

Explanation (part c)

The total pack of cards is $52$which includes hearts, spades, clubs, and diamonds. Other group includes the number of hearts in a deck which is$13$

Explanation (part d)

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case X is the number of cards that are not hearts.

The value of X is$0,1,2,3,...........,13.$

Explanation (part e)

The probability question is$P(X=13)$

Explanation (part f)

The probability so formed is

$$\begin{gathered} P(X=13)=(39,52)(38,51)(37,50)(36,49)(35,48)(34,47)(33,46)(32,45),.... \\ P(X=13)=0.0128 \end{gathered}$$

Explanation (part g)

To find the mean we are given,

$n=13,b=39,r=13$

role="math" localid="1649173640357" $$\begin{gathered} mean\left(\mu \right)=\frac{n\times r}{r+b} \\ mean\left(\mu \right)=\frac{13\times 39}{13+39} \\ mean\left(\mu \right)=9.75 \end{gathered}$$

standard deviation is ${\delta }_x=\sqrt{{\mu }_x}$

$$\begin{gathered} \delta =\sqrt{9.75} \\ \delta =1.3653 \end{gathered}$$