1. ## Hard probability problem

In the probability game, a player must spin a spinner with numbers 1-10 on it (equal chances). Based on the result of the first spinner, they have to guess if the next spinner number is going to be higher or lower. If they guess correctly they get 1 ticket. What is the probability of the player winning?

2. ## Re: Hard probability problem

Doesn't look all that "hard"- the only complicated part is the question of how this person guesses whether the next number will be higher or lower. I am going to assume that he always guesses what is, at that point, more likely. The probability that the spinner first lands on 1 is 1/10. In that case, all numbers 2, 3 4, 5, 6, 7, 8, 9, 10 are higher so the probability that the next spinner lands on a higher number is 9/10. The probability that the spinner lands on 1 and he guesses correctly (that the next spin is higher) is (1/10)(9/10)= 9/100. The probability the spinner first lands on 2 is 1/10. In that case, the probability that the next spinner lands on a higher number is 8/10. The probability that the spinner lands on 2 and he guesses correctly (that the next spin is higher) is (1/10)(8/10)= 8/100. The probability that the spinner lands on 3 is 1/10. The probability that the spinner lands on a number higher than 3 is 7/10. The probability, then, that the spinner lands on 3 and he guesses correctly (that the next spin is higher) is 7/100. The probability that the spinner lands on 4 is 1/10. The probability that the spinner lands on a number higher than 5 is 6/10. The probability, then, that the spinner lands on 4 and he guesses correctly (that the next spin is higher) is 6/100. The probability that the spinner lands on 5 is 1/10. The probability the next spin lands on a number higher is 5/10 so the probability the first spin lands on 5 and he guesses correctly (that the next spin is higher) is 5/100. The probability that the first spin lands on 6 is 1/10. The probability that the next spin lands on a number less than 6 is 5/100.

The next numbers are 4/100, 3/100, 2/100, and 1/100. The probability you seek is the product of all those numbers.

3. ## Re: Hard probability problem

If result of the first spin is 5 or less they should guess higher, and they win with prob (9/10)*0.2+(8/10)*0.2+(7/10)*0.2+(6/10)*0.2+(5/10)*0.2=0.7
If the result of the first spin is 6 or more they should guess lower and they win with the same probability as if it had been 5 or less.

So the overall probability of winning is ...

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4. ## Re: Hard probability problem

Thanks that makes more sense. The player choosing concept threw me off..