10. A coin is loaded so that the probability of heads is 0.55 and the probability of tails is 0.45, Suppose the coin is tossed twice and the results of tosses are independent. a.What is the probability of obtaining exactly two heads?
b.What is the probability of obtaining no heads?
c.What is the probability of obtaining at least one head?
two heads
tosses are independent
.55 * .55 = .3025
probability of tails = 1-.55 = .45
two tails
.45 * .45 = .2025
probability of 1 head + probability of 2 heads
= probability of head, then tail + probability of tail then head + probability of two heads
= 1 - probability of 2 tails
= 1 -.2025 = .7975
a. To find the probability of obtaining exactly two heads, we multiply the probability of getting a head on the first toss (0.55) with the probability of getting a head on the second toss (0.55) since the tosses are independent.
So, the probability of obtaining exactly two heads is:
P(Two Heads) = P(Head on first toss) * P(Head on second toss) = 0.55 * 0.55 = 0.3025
b. To find the probability of obtaining no heads, we need to find the probability of getting a tail on each toss. Since the tosses are independent, we multiply the probability of getting a tail on the first toss (0.45) with the probability of getting a tail on the second toss (0.45).
So, the probability of obtaining no heads is:
P(No Heads) = P(Tail on first toss) * P(Tail on second toss) = 0.45 * 0.45 = 0.2025
c. To find the probability of obtaining at least one head, we subtract the probability of obtaining no heads (from part b) from 1.
So, the probability of obtaining at least one head is:
P(At least one head) = 1 - P(No Heads) = 1 - 0.2025 = 0.7975
To calculate the probabilities in this scenario, we can use the concept of probability multiplication and addition rules.
a. The probability of obtaining exactly two heads can be calculated by multiplying the probabilities of getting a head on each individual toss, since the tosses are independent.
P(2 heads) = P(head) * P(head) = 0.55 * 0.55 = 0.3025
Therefore, the probability of obtaining exactly two heads is 0.3025 or 30.25%.
b. The probability of obtaining no heads can be calculated by multiplying the probabilities of getting a tail on each toss.
P(0 heads) = P(tail) * P(tail) = 0.45 * 0.45 = 0.2025
Therefore, the probability of obtaining no heads is 0.2025 or 20.25%.
c. The probability of obtaining at least one head can be calculated by taking the complement of the probability of obtaining no heads (or all tails).
P(at least one head) = 1 - P(0 heads) = 1 - 0.2025 = 0.7975
Therefore, the probability of obtaining at least one head is 0.7975 or 79.75%.