Statistics

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A total of 16 mice are sent down a maze, one by one. From previous experience, it is
believed that the probability a mouse turns right is .38

a) What is the probability that exactly 8 of these 16 mice turn right?
b) What is the probability that 8 or fewer turn right?
c) What is the probability that 8 or more turn right?
d) What is the probability that more than 3, but fewer than 10 turn right?
e) What is the probability that exactly 10 turn left?

  • Statistics -

    Because all four parts of the question involve the same set of calculations, I suggest you first work out all sixteen of the various probabilities, check that they add up to one (because if they don't, the chances are that you'll get several parts of the question wrong), and then sum up the ones you actually need. To do that, you'll need to know that:
    Prob(exactly K mice out of N turn right) = (N choose K) x (P^K) x (Q^(N-K)) where P = 0.38 and Q = 1 - P = 0.62.
    So the chances that all 16 turn right is Prob(16) = (16 choose 16) x (0.38^16) x (0.62^0) = 1.89033E-07.
    The chances that exactly 15 turn right is Prob(15) = (16 choose 15) x (0.38^15) x (0.62^1) = 16 x (0.62/0.38) x Prob(16) = 4.93476E-06.
    The chances that exactly 14 turn right is Prob(14) = (16 choose 14) x (0.38^14) x (0.62^2) = (15/2) x (0.62/0.38) x Prob(15) = 6.03858E-05.
    The chances that exactly 13 turn right is Prob(13) = (16 choose 13) x (0.38^13) x (0.62^3) = (14/3) x (0.62/0.38) x Prob(14) = 4.59780E-04.
    If you do them sequentially like this, each one can be calculated from the previous one, which makes it easy to run through the whole list quite quickly with a pocket calculator, writing them down as you go. If you're working with a spreadsheet, it doesn't make much difference whether you work each one out in full (the (N choose k) function is given by =PERMUT(N,K) in Excel) ) using the formula at the top of this post or by using the sequential method I've used here.
    Either way, the final one you work out will be:
    The chances that exactly 0 turn right is Prob(0) = (16 choose 0) x (0.38^0) x (0.62^16) = 4.76724E-04.
    If you've got them all correct they will sum to 1 - so make sure they do. You can now read off the first answer from the list of probabilities you've generated: I think it should be 1.77189E-01. If you agree with that, the answers to b) and c) will be simply a couple of easy sums, and the answer to d) can again be read straight off the list, since the probability that exactly 10 turn left is the same as the probability that exactly six turn right.

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