consider the following dice game a pair of six sided dice are rolled if the sum is equal to or l

First, we need to find the probability of winning the game (rolling a sum of 10 or more) in a si

Consider the following dice game a pair of six sided dice...

Consider the following dice game. A pair of six-sided dice is rolled. If the sum is equal to or larger than 10, then the game ends and you win 10 dollars. In all other cases, you will start over again (A pair of dice will be rolled again from scratch). Furthermore, for each rolling of a pair of dice, you pay 2 dollars. (Hint: You may think that getting a number larger than 10 is a success.) a) Find the probability that you will finish the game with more than 2 dollars of a net gain. b) Find your expected net gain obtained by playing the game. (Hint: Write the gain as a function of the number of times you roll a pair of dice.)

Transcript

00:01 So in this question you are rolling two six -sided dice, and you win if the sum is greater than or equal to 10.

00:16 So what's the probability here? well let's say that x is the sum, then the probability x is 10 is the number of ways of getting 10 divided by the number of outcomes in total, which is 36.

00:29 So for 10 we can have 4 and 6, 4 and 6, 6 and 4, or 5 and 5.

00:45 So that's 3 over 36, or 1 12th.

00:49 The probability x is 11, the number of ways of getting 11 over 36, so that's the number of 5 and 6, or 6 and 5 over 36, which is 2 over 36, or 1 in 18.

01:05 And the probability x equals 12, well in that way we can only get 6 and 6, so that's 1 in 36.

01:16 So the probability that we win on any given throw is 1 12th plus 1 18th plus 1 in 36, which is 1 6th.

01:32 So let's call that p.

01:37 Now y is the number of rolls until we win, then the probability that y is equal to y is going to be 1 minus p to the power of y minus 1 times p, as we have to fail y minus 1 times and then succeed on the yth go.

01:58 So that is 5 6ths to the y minus 1 times 1 6th, which gives us the probability y equals y is 5 to the y minus 1 over 6 to the y.

02:17 Now we need to find the probability you finish the game with more than $2 of net gain.

02:24 You are going to pay $2 for each throw and you win $10 when you win.

02:41 So that means that your winnings, so let's call z your winnings, then z is going to be 10 minus 2y.

02:54 So the probability that you finish the game with more than $2, the probability that z is greater than 2, is the probability that 10 minus 2y is greater than 2, which is the probability that 2y is less than 8, which is the probability that y is less than 4, which is the probability that y equals 1 or 2 or 3...

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