Find P(X ≥ 4) If n = 6

Kevin Katjita

Kevin Katjita

Answered question

2022-07-14

Find P(X ≥ 4) If n = 6 and P = 0.80

Answer & Explanation

alenahelenash

alenahelenash

Expert2023-06-18Added 556 answers

To find the probability P(X4), we need to calculate the cumulative probability of the random variable X being greater than or equal to 4. Given that n=6 and p=0.80, we can use the binomial cumulative distribution function to solve this.
The binomial cumulative distribution function is given by:
P(Xk)=i=kn(ni)·pi·(1p)ni
where:
- P(Xk) is the probability of X being greater than or equal to k,
- n is the total number of trials (in this case, 6),
- k is the desired minimum number of successes (in this case, 4),
- p is the probability of success in a single trial (in this case, 0.80), and
- (ni) represents the binomial coefficient, which can be calculated as (ni)=n!i!(ni)!.
Let's substitute the given values into the formula:
P(X4)=i=46(6i)·0.80i·(10.80)6i
Now we can calculate the probabilities for each value of i and sum them up:
P(X4)=(64)·0.804·(10.80)64+(65)·0.805·(10.80)65+(66)·0.806·(10.80)66
Simplifying further:
P(X4)=(64)·0.804·(10.80)2+(65)·0.805·(10.80)1+(66)·0.806·(10.80)0
Using a calculator or computer software, we can calculate the binomial coefficients:
(64)=15
(65)=6
(66)=1
Now we can substitute these values and calculate the final probability:
P(X4)=15·0.804·0.202+6·0.805·0.201+1·0.806·0.200
P(X4)0.73728
Therefore, the probability P(X4), when n=6 and p=0.80, is approximately 0.73728.

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