Question

A given binomial experiment has n=100 trials and p=1/3. Is it more likely to get x=20 successes or x=45 successes. Why?

Answer 1

Answer:

The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.

Step-by-step explanation:

Given information: n=100 and p=1/3.

According to the binomial distribution, the probability of getting r success in n trials is

$P(x=r)=^nC_rp^rq^{n-r}$

where, n is total trials, p is probability of success and q is probability of failure.

Total trials, n = 100

Probability of success, p = $\frac{1}{3}$

Probability of failure, q = $1-\frac{1}{3}=\frac{2}{3}$

The probability of 20 successes is

$P(x=20)=^{100}C_{20}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{100-20}$

$P(x=20)=\frac{100!}{20!(100-20)!}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{80}\approx 0.001257$

The probability of 45 successes is

$P(x=45)=^{100}C_{45}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{100-45}$

$P(x=45)=\frac{100!}{45!(100-45)!}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{55}\approx 0.004296$

The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.