A lottery is run in which every ticket has six numbers in a particular order. Ea

Question

A lottery is run in which every ticket has six numbers in a particular order. Each of the six numbers is in the range from 1 through 50. The person purchasing a lottery ticket can select the numbers on their ticket. On each Friday of the week, the state lottery commission holds a drawing in which six random numbers are generated. Each number in the range from 1 through 50 is equally likely.

The order of the numbers matters, so if the random numbers selected are (25, 7, 12, 37, 2, 19) then the ticket (7, 25, 12, 37, 20, 19)matches in location 3, 4, and 6. If a ticket matches in all six locations, then the ticket holder wins $100,000,000. If the ticket matches in five of the six locations, then the ticket holder wins $1,000,000. The ticket holder does not win any money for any of the other outcomes. What are the expected winnings?

Explanation / Answer

Above shown are 6 positions :

there are 2 ways in which he can win. One way to win is all 6 correct and in order. The other way is 5 positions correctly matched.

Lets first consider five positions to be matched:

Expected probability for position A to be correct is 1/50 (Out of 50 only 1 number will be correct)

Now Expected probability for position B to be correct is 1/49 (Only 1 number will be correct out of 49 remaining)

Similarly Expected probability for position C to be correct is 1/48, for D would be 1/47.

Now for winning he just need one more position to be correctly matched (E or F). For either of the position the probability of correct answer is 1/46 and if any of these positions is correct then he would win.

So Probability for him to win here is (1/50)*(1/49)*(1/48)*(1/47)*(1/46)*2

Multiplied by 2 because either E will match or F will match so 2 cases.

Lets now consider all 6 positions correctly matched case:

A, B, C and D position have expected probability as above case.

Now for E & F both to match the respective probability would be 1/46 and 1/45.

So Probability for him to win here would be (1/50)*(1/49)*(1/48)*(1/47)*(1/46)*(1/45)

Overall Winning Probability= [(1/50)*(1/49)*(1/48)*(1/47)*(1/46)*2] + [(1/50)*(1/49)*(1/48)*(1/47)*(1/46)*(1/45)]

Which comes to (91/11441304000)

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