e) A fair coin is tossed 6 times. What is the probability of getting exactly 2 heads? What is the mean of this experiment?
To find the probability of getting exactly 2 heads when a fair coin is tossed 6 times, we can use the formula for the probability of a binomial distribution.
The formula for the probability mass function of a binomial distribution is:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k) represents the probability of getting exactly k successes (in this case, exactly 2 heads)
- nCk represents the number of combinations of n items taken k at a time
- p represents the probability of success (getting a head on a fair coin toss)
- (1-p) represents the probability of failure (getting a tail on a fair coin toss)
- n represents the number of trials (in this case, 6 coin tosses)
Let's calculate the probability using these values:
P(X = 2) = (6C2) * (0.5)^2 * (1-0.5)^(6-2)
To calculate the combination value (6C2), we use the formula:
nCk = n! / (k! * (n-k)!)
Let's calculate this combination value first:
6C2 = 6! / (2! * (6-2)!)
= 6! / (2! * 4!)
= (6 * 5 * 4!) / (2! * 4!)
= (6 * 5) / 2!
= 30 / 2
= 15
Now, substitute this value into the probability formula:
P(X = 2) = 15 * (0.5)^2 * (1-0.5)^(6-2)
= 15 * 0.25 * 0.5^4
= 15 * 0.25 * 0.0625
= 0.234375
Therefore, the probability of getting exactly 2 heads when a fair coin is tossed 6 times is approximately 0.234375.
To find the mean of this experiment, we can use the formula for the mean of a binomial distribution:
Mean = n * p
where n is the number of trials and p is the probability of success.
In this case, the number of trials is 6 and the probability of success (getting a head) is 0.5.
Mean = 6 * 0.5
= 3
Therefore, the mean of this experiment is 3.