Answer:
Explanation:
The expected value for purchasing one ticket for the jackpot is the expected value of winning less the $2 cost of the ticket which is a sure cost.
The expected value of winning is the current Powerball jackpot, which is estimated to be $193 million mutiplied by the probabilities of winning.
The probability of winning is the product of the probability of choosing the five correct white balls out of the 69 white balls and the probability of choosing the one correct red ball out of the 26 red balls.
1. Probability of choosing the five correct white balls out of the 69 white balls
How many different combinations of five different balls taken from 69 balls are there?
Note that the order does not matter; so, use the combinatory formula:
[tex]Cn,x=\dfrac{n!}{x!(m-x)!}[/tex]
[tex]C(69,5)=\dfrac{69!}{5!(69-5)!}\\\\\\C(69,5)=\dfrac{69!}{5!(64)!}\\\\\\C(69,5)=11,238,513[/tex]
How many winning combinations are there? Just one
Thus the probability of winning is
[tex]\dfrac{1}{11,238,513}[/tex]
2. Probability of choosing the correct red ball out of the 26 red balls.
That is just 1 out of 26:
[tex]\dfrac{1}{26}[/tex]
3. Joint probability:
Since the two events are independent the joint probability if the product of the probabilities:
[tex]\dfrac{1}{11,238,513}\times\dfrac{1}{26}[/tex]
4. Expected value of winning
It is the value of the pot, $193 million multiplied by the joint probability
[tex]\dfrac{1}{11,238,513}\times\dfrac{1}{26}\times \$ 193,000,000=\$ 0.66[/tex]
5. Expected value for purchasing the ticket
[tex]\$ 0.66-\$2=-\$ 1.34[/tex]
It is negative meaning the the expected value is a loss of $1.34
