We know that the given binomial distribution has a 65 % rate of success. Therefore, the probability of success is p= 0.65. We want to find the probability of exactly 12 successes in 15 trials. To do so, let's first recall the Binomial Probability Formula.
Binomial Probability Formula
We have n repeated independent trials, each with a probability of success p and a probability of failure q. Also, we have that p+q=1. The binomial probability of x successes in n trials can be found by using the following formula.
P(x) = _n C_x p^x q^(n-x)
We already know that p = 0.65. Keeping in mind that p+q=1, we can calculate the value of q.
We are interested in exactly 12 successes out of 15 trials. Therefore, we can state that x= 12 and n= 15. Let's substitute all of these values into the binomial probability formula.
The probability of exactly 12 successes in 15 trials is about 0.111, or 11.1 %.
b
This time we want to find the probability that there will be at least 10 successes in 15 trials. The probability that there will be at least 10 successes is the sum of the probabilities that there are 10 or more successes.
P(at least10) =& P(10)+P(11)+P(12) +
& P(13)+ P(14)+P(15)
We will start by finding probability of exactly 10 successes. To do so, we will once again use the Binomial Probability Formula. We know that p = 0.65, q = 0.35, n= 15, and the number of successes is x= 10. We can substitute all of these values into the formula and evaluate.
The probability of 10 successes in 15 trials is about 0.212. We can find the other probabilities the same way.
Number of Successes x
P(x)= _(15) C_x (0.65)^x (0.35)^(15-x)
Simplify
10
P( 10)= _(15)C_(10) (0.65)^(10) (0.35)^(15- 10)
≈ 0.212
11
P( 11)= _(15)C_(11) (0.65)^(11) (0.35)^(15- 11)
≈ 0.179
12
P( 12)= _(15)C_(12) (0.65)^(12) (0.35)^(15- 12)
≈ 0.111
13
P( 13)= _(15)C_(13) (0.65)^(13) (0.35)^(15- 13)
≈ 0.048
14
P( 14)= _(15)C_(14) (0.65)^(14) (0.35)^(15- 14)
≈ 0.013
15
P( 15)= _(15)C_(15) (0.65)^(15) (0.35)^(15- 15)
≈ 0.002
Now, in order to find P(at least 10), we simply add all these probabilities.
P(at least10) ≈ & 0.212+0.179+
& 0.111 + 0.048+
& 0.013+0.002 = 0.565
Therefore the probability of at least 10 successes in 15 trials is about 0.565, or 56.5 %.
