3.3. Binomial Distributions and Probability http://www.ck12.org
Let’s start with a problem involving a binomial distribution.
Example A
The probability of scoring above 75% on a math test is 40%. What is the probability of scoring below 75%?
P(scoring above 75%)= 0. 40
Therefore,P(scoring below 75%)= 1 − 0. 40 = 0 .60.
Now let’s try a few problems with the binomial distribution formula.
Example B
A fair die is rolled 10 times. LetXbe the number of rolls in which we see a 2.
(a) What is the probability of seeing a 2 in any one of the rolls?
(b) What is the probability of seeing a 2 exactly once in the 10 rolls?
(a)P(X) =^16 = 0. 167
(b)
p= 0. 167
q= 1 − 0. 167 = 0. 833
n= 10
a= 1P(X=a) =nCa×pa×q(n−a)
P(X= 1 ) = 10 C 1 ×p^1 ×q(^10 −^1 )
P(X= 1 ) = 10 C 1 ×( 0. 167 )^1 ×( 0. 833 )(^10 −^1 )
P(X= 1 ) = 10 × 0. 167 × 0. 193
P(X= 1 ) = 0. 322Therefore, the probability of seeing a 2 exactly once when a die is rolled 10 times is 32.2%.
Example C
A fair die is rolled 15 times. LetXbe the number of rolls in which we see a 2.
(a) What is the probability of seeing a 2 in any one of the rolls?
(b) What is the probability of seeing a 2 exactly twice in the 15 rolls?
(a)P(X) =^16 = 0. 167
(b)
p= 0. 167
q= 1 − 0. 167 = 0. 833
n= 15
a= 2