A club runs a dice game at its open day…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A club runs a dice game at its open day. A fair die is thrown once, and the club pays the player the square of the number shown, in rupees. A throw of pays rupees, for example. The treasurer wants three numbers before she fixes the entry fee. First, the fair entry fee, which is the fee that makes the expected net gain zero. Second, the variance of the payout. Third, the standard deviation of the payout. A large variance means that the club could lose a lot of money on a busy day, even if the fee is fair. The treasurer will compare the standard deviation with the fee to judge how risky the game is. Give decimals to two places.
Need a reminder? The key ideaShow
For a distribution, and with . For , the mean is and the variance is .
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The fair fee. rupees.
Step 2. The mean of the fourth powers. .
Step 3. The variance. .
Step 4. The standard deviation. rupees.
Answer: about rupees, and about rupees.
What it means. The standard deviation is about of the fee. A single game swings widely: a throw of pays rupee and a throw of pays rupees. Only over many games does the average settle near the fee.
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