4. A fair die is rolled 150 times. What is the probability of a 6 coming up between 20 and 30 times? (20 times or 30 times would count.)
We could do this by constructing the probability distribution as in the previous problem, but that would take so much effort that it would be worthwhile to use the normal approximation to the binomial distribution.
The number of times the experiment is repeated is
The probability of success each time is
The mean is
np = 150(1/6) = 25
and the standard deviation is
Since we are counting the 20 and the 30, the upper limit is 30.5 and the lower limit is 19.5.
We convert these numbers to z-scores using the formula
If we look up a z-score of 1.205 in the bell shaped curve table in the back of the book, 1.205 is halfway between 1.20 and 1.21, so we use the area which is halfway between the areas for these two z-scores. This gives us an area of .3849. That is the area under the bell shaped curve between the mean and 1.205 standard deviations above the mean. The area under the bell shaped curve between the mean and 1.205 standard deviations below the mean will also be .3849, since the bell shaped curve is symmetric. The total area is
.3849 + .3849 = .7698
and this is the probability of rolling between 20 and 30 6's.