Final answer:
The probability of getting 5 heads in 5 flips when the coins are put back after each flip is 3.125%, and the probability of getting 5 heads in 5 flips without replacement is 16.25%.
Step-by-step explanation:
The probability of getting 5 heads in 5 flips when the coins are put back after each flip can be calculated using the binomial probability formula.
The formula is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of flips, k is the number of heads, and p is the probability of getting a head on each flip. In this case, n=5, k=5, and p=0.5.
Plugging these values into the formula gives:
P(X=5) = C(5, 5) * 0.5^5 * (1-0.5)^(5-5) = 1 * 0.5^5 * 0.5^0 = 0.5^5 = 0.03125, or 3.125%
When the coins are not put back after each flip, the probability changes because the outcome of each flip affects the subsequent flips.
To calculate the probability of getting 5 heads in 5 flips without replacement, we need to use a different approach.
Since there is only one double-headed coin, the probability of selecting it is 1/10, and the probability of selecting a fair coin is 9/10.
Once a coin is selected, the probability of getting a head on that coin is 1 for the double-headed coin and 0.5 for the fair coin.
We can calculate the probability of getting 5 heads in 5 flips without replacement by considering all possible ways of selecting the coins:
P(5 heads in 5 flips) = P(1 double-headed coin, 5 heads) + P(2 double-headed coins, 3 heads) + P(3 double-headed coins, 1 head)
P(5 heads in 5 flips) = (1/10) * (1^5) * (9/10)^4 + (1/10) * (1^3) * (9/10)^2 + (1/10) * (1^1) * (9/10)^0 = 0.1625, or 16.25%