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4.3 THE BINOMIAL DISTRIBUTION 105
Solution: This probability is an entry in the table. No addition or subtraction is necessary.
P1X … 52 = .9666.
(b) Six or more will be color blind.
Solution:
We cannot find this probability directly in the table. To find the answer, we
use the concept of complementary probabilities. The probability that six or
more are color blind is the complement of the probability that five or fewer
are color blind. That is, this set is the complement of the set specified in
part a; therefore,
P1X Ú 62 = 1 - P1X … 52 = 1 - .9666 = .0334
(c) Between six and nine inclusive will be color blind.
Solution: We find this by subtracting the probability that X is less than or equal to 5
from the probability that X is less than or equal to 9. That is,
P16 … X … 92 = P1X … 92 - P1X … 52 = .9999 - .9666 = .0333
(d) Two, three, or four will be color blind.
Solution:
This is the probability that X is between 2 and 4 inclusive.
P12 … X … 42 = P1X … 42 - P1X … 12 = .9020 - .2712 = .6308
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Using Table B When p>.5 Table B does not give probabilities for values of
p greater than .5. We may obtain probabilities from Table B, however, by restating the
problem in terms of the probability of a failure, 1 - p, rather than in terms of the probability
of a success, p. As part of the restatement, we must also think in terms of the number
of failures, n - x, rather than the number of successes, x. We may summarize this
idea as follows:
P1X = x ƒ n, p 7 .502 = P1X = n - x ƒ n,1 - p2
(4.3.3)
In words, Equation 4.3.3 says, “The probability that X is equal to some specified value given
the sample size and a probability of success greater than .5 is equal to the probability that
X is equal to n - x given the sample size and the probability of a failure of 1 - p. ” For
purposes of using the binomial table we treat the probability of a failure as though it were
the probability of a success. When p is greater than .5, we may obtain cumulative probabilities
from Table B by using the following relationship:
P1X … x ƒ n, p 7 .502 = P1X Ú n - x ƒ n,1 - p2
(4.3.4)
Finally, to use Table B to find the probability that X is greater than or equal to some x
when P 7 .5, we use the following relationship:
P1X Ú x ƒ n, p 7 .502 = P1X … n - x ƒ n,1 - p2
(4.3.5)