To find the probability that the 6th tose out of 10 tosses will be heads. Given there are exactly 2 heads out of 10 tosses. As in precending segment continue to assume all tosses are independent and that each toss has same fixed probability of heads. Answer is 1/5
To find the probability that the 6th toss out of 10 tosses will be heads, given that there are exactly 2 heads out of 10 tosses, we first need to calculate the probability of getting exactly 2 heads in 10 tosses.
The probability of getting exactly 2 heads in 10 tosses can be calculated using the binomial probability formula:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- n is the number of trials (10 tosses)
- k is the number of successful outcomes (2 heads)
- p is the probability of success on each trial (probability of getting heads)
Using the binomial probability formula:
P(X=2) = (10 choose 2) * p^2 * (1-p)^8
Given that there are exactly 2 heads out of 10 tosses, the probability of getting exactly 2 heads is:
P(X=2) = C(10,2) * p^2 * (1-p)^8
Now, we know that there are exactly 2 heads in 10 tosses, which means that there are 8 tails. Therefore, the probability of getting tails on each of the remaining 8 tosses is (1-p)^8.
Since we want to find the probability that the 6th toss out of 10 tosses will be heads, given that there are exactly 2 heads out of 10 tosses, we need to find the probability that the 6th toss will be heads when there are exactly 2 heads and 8 tails in 10 tosses.
Since the probability of getting heads or tails on each toss is independent and has the same fixed probability, the probability of the 6th toss being heads given the conditions is the same as the probability of getting heads in 10 tosses.
Therefore, the probability that the 6th toss out of 10 tosses will be heads, given that there are exactly 2 heads out of 10 tosses, is the same as the probability of getting heads in 10 tosses.
Since we already calculated the probability of getting 2 heads out of 10 tosses earlier, we can use that result:
P(X=2) = C(10,2) * p^2 * (1-p)^8
Since there are exactly 2 heads, the probability of getting heads is p^2.
Therefore, the probability of the 6th toss being heads when there are exactly 2 heads and 8 tails in 10 tosses is:
P(6th toss is heads | 2 heads out of 10 tosses) = p^2 = 1/5
So, the probability that the 6th toss out of 10 tosses will be heads, given that there are exactly 2 heads out of 10 tosses, is 1/5.