Consider a binomial experiment with 20 trials and probability 0.45 on a single trial. Use the binomial distribution to find the probability of exactly 10 successes. Round your answer to the thousandths place
Use the binomial probability function:
P(x) = nCx(p^x)[q^(n-x)]
n = 20
x = 10
p = 0.45
q = 1 - p = 0.55
Substitute into the function and go from there.
Note:
You can also use a binomial probability function table with the values listed to find the probability as well. It is an easier way of doing this problem.
I hope this will help get you started.
THis answer is wrong
To find the probability of exactly 10 successes in a binomial experiment with 20 trials and a probability of 0.45 on a single trial, we can use the binomial distribution formula.
The formula for the probability of k successes in n trials with probability p of success on a single trial is:
P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))
where:
- n is the number of trials
- k is the number of successes
- p is the probability of success on a single trial
- nCk is the number of combinations of n items taken k at a time
Now, let's substitute the given values into the formula:
P(X = 10) = (20C10) * (0.45^10) * ((1-0.45)^(20-10))
Calculating each part separately:
(20C10) = (20!)/((10!)*(20-10)!)
= (20!)/(10!*10!)
= (20*19*18*17*16*15*14*13*12*11)/(10*9*8*7*6*5*4*3*2*1)
≈ 184,748
(0.45^10) ≈ 0.0133
((1-0.45)^(20-10)) = (0.55^10) ≈ 0.0535
Now, let's calculate the final probability by multiplying all the parts together:
P(X = 10) ≈ (184,748) * (0.0133) * (0.0535)
≈ 131.306
Therefore, the probability of exactly 10 successes in a binomial experiment with 20 trials and a probability of 0.45 on a single trial is approximately 0.131 (rounded to the thousandths place).
To find the probability of exactly 10 successes in a binomial experiment with 20 trials and a probability of success of 0.45 on a single trial, we can use the binomial distribution formula.
The formula for the binomial distribution probability mass function is:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
where:
- P(X=k) is the probability of getting exactly k successes
- n is the number of trials
- p is the probability of success on a single trial
- (n C k) is the binomial coefficient, or the number of ways to choose k successes from n trials
Plugging in the values:
- n = 20 (the number of trials)
- p = 0.45 (the probability of success on a single trial)
- k = 10 (the number of successes we are interested in)
First, we need to calculate the binomial coefficient (n C k). The binomial coefficient can be calculated using the formula:
(n C k) = n! / (k! * (n-k)!)
Using the values n = 20 and k = 10:
(20 C 10) = 20! / (10! * (20-10)!)
Now, let's calculate the probability:
P(X=10) = (20 C 10) * (0.45)^10 * (1-0.45)^(20-10)
(20 C 10) = 20! / (10! * 10!)
(20 C 10) = 20! / (10! * 10!)
P(X=10) = (20 C 10) * (0.45)^10 * (1-0.45)^(20-10)
Calculating the binomial coefficient and simplifying:
(20 C 10) = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11) / (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
(20 C 10) ≈ 184, 756
Now, plug in the values into the probability formula and round the answer to the thousandths place:
P(X=10) = (184, 756) * (0.45)^10 * (1-0.45)^(20-10)
P(X=10) ≈ 0.005