Question

if you draw two spades on the two consecutive draws with replacement from standard deck of cards you win $23. otherwise pay me $5. find the expected values of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

Answer 1

Given a deck of card, the sample space (T) is 52, there 13 spades (S), therefore the probability of drawing two spades on two cosecutive draws is given below

$\begin{gathered} n(T)=52 \\ n(S)=13 \\ Pr(S)=\frac{13}{52}=\frac{1}{4} \\ \text{ Since there is replacement} \\ Pr(S)\text{ and Pr(S) =}\frac{1}{4}\times\frac{1}{4}=\frac{1}{16} \end{gathered}$

The probability of not picking two consecutive spades will be

$\begin{gathered} Pr(S_1S_2)^{\prime}=1-Pr(S_1S_2) \\ =1-\frac{1}{16}=\frac{15}{16} \end{gathered}$

The expected value for drawing two consecutive spades will be

$\begin{gathered} ExpectedValue=xP(x)_{} \\ x=23,P(x)=\frac{1}{16}_{} \\ ExpectedValue=23\times\frac{1}{16}=\frac{23}{16}=1.44 \end{gathered}$

The expected value for not drawing two consecutive spades will be