Question

A person bets 1 dollar to b dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. Find b so that the bet is fair.

Answer 1

Answer:

$3.25

Step-by-step explanation:

In an ordinary deck of cards, we will notice that there are 52 cards.

However, we have 4 suits and 13 cards in each one making a total of 52, right?.

So, the probability that a person draws two cards and they will be of the same   suit is:

$P(X) =\dfrac{ \bigg (^4_1 \bigg) \times \bigg ( ^{13}_{2}\bigg) }{ \bigg ( ^{52}_{2} \bigg)}$

$P(X) =\dfrac{ \dfrac{4!}{1!(4-1)!} \times \dfrac{13!}{2!(13-2)!} }{ \dfrac{52!}{2!(52-2)!} }$

$P(X) =\dfrac{4}{17}$

Given that; A person bets 1 dollar to b dollars;

To make the bet fair is;

$\dfrac{4}{17}b - \dfrac{13}{17}(1) = 0$

$= \dfrac{4}{17}b = \dfrac{13}{17}$

multiply both sides by 17

4b = 13

b = 13/4

b = 3.25

Therefore, to make the fair, the value of b need to be $3.25

Answer 2

Using the expected value formular, the value of b such that the game is fair is 3.25

Total number of cards in a deck = 52

Number of cards per suit = 13

Number of suits = 4

Selection without replacement :

• Club = C ; Spade = S ; Heart = H ; Diamonds = D

(C and C) or (H and H) or (S and S) or (D and D)

[(13/52 × 12/51) + (13/52 × 12/51) + (13/52 × 12/51) + (13/52 × 12/51)] = 0.2353

Probability of not selecting a card of the same suit :

• 1 - 0.2353 = 0.7647

For game to be fair :

• Σ[(X × P(X)] = 0

X : ______ b _____ - 1

P(X): __ 0.2353__ 0.7647

[(0.2353b + (-1 × 0.7647)] = 0

0.2353b - 0.7647 = 0

0.2353b = 0.7647

Divide both sides by 0.2353

b = 0.7647 / 0.2353

b = 3.249

b = 3.25

Hence, the value of b is 3.25

Learn more : https://brainly.com/question/19755846