In a lottery game, a player picks shox numbers from 1 to 20. If the player matches all six numbers, they win 20,000 dollars. Otherwise, they lose $1. What is the expected value of this game? $​

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$0.8514 per play.

Step-by-step explanation:

To start solving the problem, we need to understand the definitions of words, the most important of which is "expected value".

Wiki defines expected value as:

"The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment. More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values."

To mathematically calculate the expected value, we will need to do the sum of the product

x*P(x)

for all possible (outcome) values of x, that is, all values of x for which the probability is non-zero.

Here the possible outcomes are x=-1 (lose) or x=20000 (win)

Total possible number of outcomes = C(24,6) [24 choose 6]

=24!/(6!18!)

= 134596

Out of which there is only one winning combination.

Therefore we conclude:

P(win 20000)=1/134596

P(lose 1)=134595/134596

and hence the expected value is:

20000*(1/134596)+(-1)*(134595/134596)

=-114595/134596

=-0.8514 (rounded to four places after decimal)

Interpretation:

In the long run, if the game is played many, many times, the average loss is $0.8514 per play.