A trick coin has been weighted so that heads occurs
with a probability of p � �2
3�, and p(tails) � �1
3�. If you
toss this coin 72 times,
a. How many heads would you expect to get on
average?
b. What is the probability of getting more than 50
heads?
c. What is the probability of getting exactly 50 heads?
To answer these questions, we can use the concept of expected value and probability in statistics. Let's break down each question and explain how to find the answer.
a. Expected number of heads:
The expected value is calculated by multiplying each possible outcome by its probability and summing them up. In this case, since heads occurs with a probability of 2/3, we can calculate the expected number of heads as follows:
Expected number of heads = (probability of heads) * (number of tosses)
Expected number of heads = (2/3) * 72
Expected number of heads = 48
Therefore, on average, you would expect to get 48 heads.
b. Probability of getting more than 50 heads:
To find the probability of getting more than 50 heads, you can use the binomial probability distribution formula. However, since calculating it manually can be quite involved, we can use statistical software or an online calculator to find the exact probability easily.
An alternative approach is to use the normal approximation to the binomial distribution when the sample size is large. In this case, we can use the normal distribution to estimate the probability.
First, we need to calculate the mean and standard deviation. The mean (μ) is equal to the expected value, which is 48. The standard deviation (σ) can be calculated using the formula for the standard deviation of a binomial distribution:
Standard deviation = sqrt((number of tosses) * (probability of success) * (probability of failure))
Standard deviation = sqrt(72 * (2/3) * (1/3))
Standard deviation ≈ 4.27
Now, we can use the normal distribution to estimate the probability of getting more than 50 heads. By calculating the z-score and looking it up in a standard normal distribution table or using a calculator, we can find the probability.
P(X > 50) = P(Z > (50 - μ) / σ)
P(X > 50) = P(Z > (50 - 48) / 4.27)
Using a standard normal distribution table or calculator, you can find the probability of getting more than 50 heads.
c. Probability of getting exactly 50 heads:
To find the probability of getting exactly 50 heads, we can use the binomial probability formula:
P(X = k) = (nCk) * (p^k) * ((1 - p)^(n - k))
where n is the number of trials (in this case, 72), k is the number of successful outcomes (50 in this case), p is the probability of success (2/3), and (nCk) is the binomial coefficient which represents the number of ways to choose k successes from n trials.
Using this formula, you can calculate the probability of getting exactly 50 heads.