Answer:
a) 5/16
b) 31/32
Step-by-step explanation:
A coin toss is an example of a binomial distribution, where the following conditions are met:
- A discrete and fixed number of trials, n
- Trials are independent of each other
- Two possible outcomes: success and failure
- Probability of success, p, is constant
In this case, there are 5 coin tosses, so n = 5. There are two outcomes, heads (success) and tails (failure). Each coin toss is independent, and the probability of success is always p = 0.5.
The probability of getting exactly r number of successes out of n trials is:
P = nCr pʳ (1−p)ⁿ⁻ʳ
where
- nCr is the number of combinations of r successes from n trials
- p is the probability of success
- r is the number of successes
- 1−p is the probability of failure
- n−r is the number of failures
Since n = 5 and p = 0.5, this equation becomes:
P = 5Cr (0.5)ʳ (1−0.5)⁵⁻ʳ
P = 5Cr (0.5)ʳ (0.5)⁵⁻ʳ
P = 5Cr (0.5)⁵
a) If r = 3, then the probability is:
P = ₅C₃ (0.5)⁵
P = 10 (0.5)⁵
P = 5/16
b) The probability of at least one head is the probability of one head, two heads, three heads, four heads, or five heads.
P(at least 1) = P(1) + P(2) + P(3) + P(4) + P(5)
Or, we can simply say it is 1 minus the probability of no heads.
P(at least 1) = 1 − P(0)
P(at least 1) = 1 − ₅C₀ (0.5)⁵
P(at least 1) = 1 − 1 (0.5)⁵
P(at least 1) = 31/32
