A sorcerer has hidden an M&M in one of an infinite row of boxes. You need to fin

Question

A sorcerer has hidden an M&M in one of an infinite row of boxes. You need to find it as quickly as possible. As a matter of fact, you only have time to look in 11 boxes. The sorcerer says that he will give you a hint, by tossing a coin 100 times, and counting the number of heads, all in a flash. He will then tell you the sum of the number that the box is in, and the number of heads that he got out of 100 tosses. If the sum is 75, you decide to look in boxes 20 through 30. What is the chance that you will locate the M&M? Choose the closest answer. (73%, 68%, 70%)

Explanation / Answer

Here the sum is 75. Here let say the box number is X.

so X + Y = 75 ; where Y is the total number of heads out of 100 coin toss.

By normal approximation of Y,

E(Y) = 50

Var(Y) = 0.5 * 0.5 * 100 = 25

STDEV(Y) = 5

Now we look into boxes 20 through 30 that means value of Y can vary from 45 to 55.

Pr(45 Y 55) = NORM (45 Y 55; 50 ; 5)

By continuity correction

Pr(45 Y 55) = NORM (45 Y 55; 50 ; 5) = Pr(Y<  55.5 ; 50 ; 5) - Pr(Y < 44.5; 50; 5) = (Z2 ) - (Z1 )

Here is the normal standard cumulative distriubtion.

Z2 = (55.5 - 50)/ 5 = 1.1 ; Z1 = (44.5 - 50)/ 5 = -1.1

(Z2 ) - (Z1 ) =  (1.1) - (-1.1)  

from Z - table

(1.1) - (-1.1) = 0.8643 - 0.1357 = 0.7286 or say 73%

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