Question

3. Suppose that you draw two cards from a standard 52 card deck. You replace the first card and shuffle thoroughly before you draw the second. [p-4-9-45] a. What is the probability that the first card is 7? b. What is the probability that both cards are 7? c. What is the probability that the first card is not a 7? d. What is the probability that both cards are not 7?

Answer 1

`\text{a)} (1)/(17) \text{b)} (1)/(221)\text{c)}(12)/(13)\text{d)}(188)/(221)`

Explanation:

GIVEN: You draw two cards from a standard
`52` card deck. You replace the first card and shuffle thoroughly before you draw the second.

TO FIND: a) What is the probability that the first card is
`7` b) What is the probability that both cards are

SOLUTION:

Let
`\text{A}` and
`\text{B}` be two events such that

`\text{A}`
`=\text{first card drawn is 7}`

`\text{B}`
`=\text{Second card is 7}`

`\text{probability that first card is 7 =P(A)}`

`\text{P(A)}=\frac{\text{total 7 numbered cards}}{\text{total cards in deck}}`

`\text{P(A)}=`
`(4)/(52)=(1)/(13)`

a)

`\text{probability that first card is 7 =P(A)}`

`\text{P(A)}=(1)/(13)`

b)

`\text{probability that both cards are 7}=`
`\text{first card is 7}*\text{second card is 7}`

`\text{probability that both cards are 7}=`
`\text{P(A).P(B)}`

`\text{P(B)}=\frac{\text{total number of 7 numbered cards left}}{\text{total number of cards in deck left}}`

`\text{P(B)}=(3)/(51)=(1)/(17)`

`\text{probability that both cards are 7}=`
`(1)/(13)*(1)/(17)`

`\text{probability that both cards are 7}=`
`(1)/(221)`

c)

`\text{probability that first card is not 7}=1-\text{probability that first card is 7}`

`\text{probability that first card is not 7}=`
`1-\text{P(A)}`

`\text{probability that first card is not 7}=`
`1-(1)/(13)`

`\text{probability that first card is not 7}=`
`(12)/(13)`

d)

`\text{probability that both cards are not 7}=`
`(\text{first card is not 7}).(\text{second card is not 7})`

`\text{probability that both cards are not 7}=`
`\text{[1-P(A)].[1-P(B)]}`

`\text{P(B)}=\frac{\text{total 7 numbered cards}}{\text{total cards left in deck}}`

`\text{probability that both cards are not 7}=`
`(1-(1)/(13))*(1-(4)/(51))`

`\text{probability that both cards are not 7}=`
`(12)/(13)*(47)/(51)`