We expect to lose $0.37 per lottery ticket
<u>Explanation:</u>
six winning numbers from = { 1, 2, 3, ....., 50}
So, the probability of winning:
![P(win) = \frac{ no of favorable outcomes}{no of possible outcomes}](https://tex.z-dn.net/?f=P%28win%29%20%3D%20%5Cfrac%7B%20no%20of%20favorable%20outcomes%7D%7Bno%20of%20possible%20outcomes%7D)
![P(win) = \frac{1}{^5^0C_6} \\\\P (win) = \frac{6! X (50 - 6)!}{50!} \\\\P(win) = \frac{6! X 44!}{50!} \\\\P(win) = \frac{1}{15,890,700}](https://tex.z-dn.net/?f=P%28win%29%20%3D%20%5Cfrac%7B1%7D%7B%5E5%5E0C_6%7D%20%5C%5C%5C%5CP%20%28win%29%20%3D%20%5Cfrac%7B6%21%20X%20%2850%20-%206%29%21%7D%7B50%21%7D%20%5C%5C%5C%5CP%28win%29%20%3D%20%5Cfrac%7B6%21%20X%2044%21%7D%7B50%21%7D%20%5C%5C%5C%5CP%28win%29%20%3D%20%5Cfrac%7B1%7D%7B15%2C890%2C700%7D)
The probability of losing would be:
P(loss) = 1 - P(win)
![P(loss) = 1 - \frac{1}{15,890,700} \\\\P(loss) = \frac{15,890,699}{15,890,700}](https://tex.z-dn.net/?f=P%28loss%29%20%3D%201%20-%20%5Cfrac%7B1%7D%7B15%2C890%2C700%7D%20%5C%5C%5C%5CP%28loss%29%20%3D%20%5Cfrac%7B15%2C890%2C699%7D%7B15%2C890%2C700%7D)
According to the question,
When we win, then we gain $10 million and lose the cost of the lottery ticket.
So,
$10,000,000 - 1 = $9,999,999
When we lose, then we lose the cost of the lottery ticket = $1
The expected value is the sum of the product of each possibility x with its probability P(x):
E(x) = ∑ xP(x)
![= 9,999,999 X \frac{1}{15,890,700} + ( -1 ) X \frac{15,890,699}{15,890,700} \\\\=- \frac{5,890,700}{15,890,700} \\\\= - \frac{58,907}{158,907} \\\\= - 0.37](https://tex.z-dn.net/?f=%3D%209%2C999%2C999%20X%20%5Cfrac%7B1%7D%7B15%2C890%2C700%7D%20%20%2B%20%28%20-1%20%29%20X%20%5Cfrac%7B15%2C890%2C699%7D%7B15%2C890%2C700%7D%20%5C%5C%5C%5C%3D-%20%5Cfrac%7B5%2C890%2C700%7D%7B15%2C890%2C700%7D%20%5C%5C%5C%5C%3D%20-%20%5Cfrac%7B58%2C907%7D%7B158%2C907%7D%20%5C%5C%5C%5C%3D%20-%200.37)
Thus, we expect to lose $0.37 per lottery ticket