## Question 418:

1

I think the binomial distribution can help you here. This will take into account both the probability of the event, which you've stated as 1/10 or .1 for picking a number from 1 to 10.  Lets run through some scenarios and review the formula.

If you have access to Excel, it will make your calculations a lot faster. Use the formula =BINOMDIST(). The parameters you use are the number of correct guesses, the total guesses, the probability of a correct guess and either the cumulative probability of exact probability. We're using the number of guesses as equal to the number of people making the guess.

For the 1st example of 5 people, the probability of at least 1 person guessing 4 is equal to the probability of 1,2,3,4 and all 5 people guessing, which is the same as the probability of 1 minus the probability of 0 people guessing. To me the easiest thing to do would be find this probability, then subtract it from 1.  The excel formula would be =BINOMDIST(0,5,0.1,FALSE), which provides us with .5905, therefore the probability of more than 0 getting it right is 1-.5905 = .4095. So your original question has the answer of .5, which is close but not correct (not sure if you just added up .1*5, but the events are independent so we can't add them here).

The chance of exactly two out of 5 people guessing a 4 (no more or no less) is just =BINOMDIST(2,5,0.1,FALSE) = .0729 or about a 7.3% chance of that.

So this should answer your subsequent questions. Just change the values in the formula for the larger sample, assuming they are still guessing a number from 1-10.

• 2 out of 8 people guessing a 4 =BINOMDIST(2,8,0.1,FALSE) = .148
• 2 out of 20 people guessing a 4 =BINOMDIST(2,20,0.1,FALSE) = .285
•  3 out of 50 people guessing a 4 =BINOMDIST(3,20,0.1,FALSE) = .190

See question 175 for a more detailed walk through of the binomial probability formula.