Simple question about chance

profitius

If you have 4 people and each have a 1 in 4 chance of winning a prize, does that mean theres an even chance that the prize will be won. The obvious answer is evens chance but maths isn't always obvious.

hivizman

profitius wrote: »

If you have 4 people and each have a 1 in 4 chance of winning a prize, does that mean theres an even chance that the prize will be won. The obvious answer is evens chance but maths isn't always obvious.

I'd be interested to read why you believe that the "obvious" answer is that there is an even chance (that is, one chance in 2 or 50%) that the prize will be won.

In fact, you haven't provided enough information to give an unambiguous answer. Two of the various possible scenarios are the following:

Scenario 1

Each person has an equal chance of winning the prize. In this case, the probability that any one person will win the prize is 1 in 4 (25%), and someone (and only one person) must win the prize.

An example of this would be where the winner of the prize is determined by drawing one card at random from a deck of 52 standard playing cards. If a spade is drawn, person A wins, if a heart is drawn, person B wins, if a diamond is drawn, person C wins, and if a club is drawn, person D wins. As there are 13 cards in each suit, the probability of drawing a particular suit is 13 in 52 or 1 in 4. The card must be a spade, heart, diamond or club, so someone must win.

Scenario 2

Each person has a 1 in 4 chance of winning a prize, independent of the outcome for all the other players. In this situation, more than one player could win a prize, or no player could win a prize.

The probabilities that n persons will win a prize are given by the binomial distribution, as follows (rounded to 3dp):

0 0.316
1 0.422
2 0.211
3 0.047
4 0.004

So the probability that at least one prize will be won is 0.684 (slightly more than two thirds).

It's important to note the assumption that the outcome for each person is independent of the outcome for each other player.

An example of this sort of situation would be where a prize is awarded for attaining a particular level of achievement (for example, exceeding 2 metres in the high jump). If each of the four players can jump over 2m one time in four on average, then the probability that at least one of them jumps over 2m and wins a prize is 0.684.

In practice, the probabilities may not be independent. For example, if the first player achieves a 2m high jump, then this may encourage the later players to perform better than their past achievements would predict.

profitius

hivizman wrote: »

I'd be interested to read why you believe that the "obvious" answer is that there is an even chance (that is, one chance in 2 or 50%) that the prize will be won.

In fact, you haven't provided enough information to give an unambiguous answer. Two of the various possible scenarios are the following:

Scenario 1

Each person has an equal chance of winning the prize. In this case, the probability that any one person will win the prize is 1 in 4 (25%), and someone (and only one person) must win the prize.

An example of this would be where the winner of the prize is determined by drawing one card at random from a deck of 52 standard playing cards. If a spade is drawn, person A wins, if a heart is drawn, person B wins, if a diamond is drawn, person C wins, and if a club is drawn, person D wins. As there are 13 cards in each suit, the probability of drawing a particular suit is 13 in 52 or 1 in 4. The card must be a spade, heart, diamond or club, so someone must win.

Scenario 2

Each person has a 1 in 4 chance of winning a prize, independent of the outcome for all the other players. In this situation, more than one player could win a prize, or no player could win a prize.

The probabilities that n persons will win a prize are given by the binomial distribution, as follows (rounded to 3dp):

0 0.316
1 0.422
2 0.211
3 0.047
4 0.004

So the probability that at least one prize will be won is 0.684 (slightly more than two thirds).

It's important to note the assumption that the outcome for each person is independent of the outcome for each other player.

An example of this sort of situation would be where a prize is awarded for attaining a particular level of achievement (for example, exceeding 2 metres in the high jump). If each of the four players can jump over 2m one time in four on average, then the probability that at least one of them jumps over 2m and wins a prize is 0.684.

In practice, the probabilities may not be independent. For example, if the first player achieves a 2m high jump, then this may encourage the later players to perform better than their past achievements would predict.

Cheers.

It was scenario B. All independant and there could be 4 winners or no winner. Most people say its an evens chance because they add the 4 1 in 4 chances and get 1. I asked the question because that was the first thing that popped into my head until I thought about it.