Difference between betting X numbers in one wheel-of-luck turn and betting 1 number in X wheel-of-luck turns

Question

The scenario is simple:

A wheel of luck has 10 numbers and people can bet on any number and win a chocolate if the wheel spins in their number. What is the difference between betting for 5 numbers in a single turn and betting 1 number in 5 turns? What is the formula for deciding this?

Answer 1

The probability of success in the case in which you bet $5$ numbers is easy to compute: it's $\frac12$, since there are $10$ numbers and $\frac5{10}=\frac12$.

Now, if you bet one number five times, how can you win? There are thesse possibilities:

• You can win every single time. The odds are $\left(\frac1{10}\right)^5$.
• You can win exactly four times. The odds are $5\times\left(\frac1{10}\right)^4\times\left(1-\frac1{10}\right)$.
• You can win exactly three times. The odds are $10\times\left(\frac1{10}\right)^3\times\left(1-\frac1{10}\right)^2$.
• You can win exactly two times. The odds are $10\times\left(\frac1{10}\right)^2\times\left(1-\frac1{10}\right)^3$.
• You can win exactly once. The odds are $5\times\frac1{10}\times\left(1-\frac1{10}\right)^4$.

So, the probability of success is the sum of these numbers: $0.41761$.

Answer 2

There is a difference between average payoff which is $50$% in either case and how often you will have a winner. As indicated by Santos, one bet on five numbers will lead to a win $50$% of the time while betting one number five times leads to one or more wins $42$% of the time.

Another way of looking at it is the standard deviation of the payoff is smaller when you bet on five numbers at one time.