Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(X9) , n=10 , p=0.8
The Correct Answer and Explanation is:
To find the probability P(X=9)P(X = 9)P(X=9) for a binomial distribution, we use the binomial probability formula:P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn)pk(1−p)n−k
Where:
- nnn is the number of trials,
- kkk is the number of successes,
- ppp is the probability of success on a single trial,
- (nk)\binom{n}{k}(kn) is the binomial coefficient, also known as “n choose k”.
Given:
- n=10n = 10n=10 (the number of trials),
- p=0.8p = 0.8p=0.8 (the probability of success on a single trial),
- k=9k = 9k=9 (the number of successes we are interested in).
Let’s break this down:
Step 1: Find the binomial coefficient
The binomial coefficient (nk)\binom{n}{k}(kn) is calculated as:(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}(kn)=k!(n−k)!n!
For n=10n = 10n=10 and k=9k = 9k=9:(109)=10!9!(10−9)!=10!9!1!=10×9!9!×1!=10\binom{10}{9} = \frac{10!}{9!(10-9)!} = \frac{10!}{9!1!} = \frac{10 \times 9!}{9! \times 1!} = 10(910)=9!(10−9)!10!=9!1!10!=9!×1!10×9!=10
Step 2: Apply the formula
Now, we can apply the binomial probability formula:P(X=9)=(109)(0.8)9(1−0.8)10−9P(X = 9) = \binom{10}{9} (0.8)^9 (1 – 0.8)^{10 – 9}P(X=9)=(910)(0.8)9(1−0.8)10−9P(X=9)=10×(0.8)9×(0.2)1P(X = 9) = 10 \times (0.8)^9 \times (0.2)^1P(X=9)=10×(0.8)9×(0.2)1
Step 3: Simplify the powers
- (0.8)9≈0.1342(0.8)^9 \approx 0.1342(0.8)9≈0.1342
- (0.2)1=0.2(0.2)^1 = 0.2(0.2)1=0.2
So:P(X=9)=10×0.1342×0.2P(X = 9) = 10 \times 0.1342 \times 0.2P(X=9)=10×0.1342×0.2P(X=9)=10×0.02684=0.2684P(X = 9) = 10 \times 0.02684 = 0.2684P(X=9)=10×0.02684=0.2684
Final Answer:
The probability P(X=9)P(X = 9)P(X=9) is approximately 0.2684.
This means that the probability of getting exactly 9 successes in 10 trials with a success probability of 0.8 is about 0.2684, or 26.84%.
