A balanced coin is tossed 8 times. What is the probability of getting exactly 2 heads given that there were at least 2 heads in the 8 tosses?

Nine coins are tossed; in how many ways can you have at least two tails?

four coinsare tossed hat is the probability that atleast 2 tails turn up

To find the probability of getting exactly 2 heads given that there were at least 2 heads in the 8 tosses, we need to determine the total number of outcomes where there are at least 2 heads in the 8 tosses, and then calculate the number of outcomes where exactly 2 of those tosses result in heads.

Let's break down the calculation step by step:

Step 1: Determine the total number of outcomes where there are at least 2 heads in the 8 tosses.
To find this, we need to identify the complementary event, which is the event where there are fewer than 2 heads in the 8 tosses. In other words, we need to find the probability of getting 0 or 1 head in the 8 tosses and subtract it from 1 (the total probability).

Number of outcomes with 0 heads = C(8, 0) = 1 (since there is only 1 way to get all tails)
Number of outcomes with 1 head = C(8, 1) = 8 (since there are 8 ways to get only 1 head)

Total number of outcomes with fewer than 2 heads = 1 + 8 = 9

So, the total number of outcomes where there are at least 2 heads in the 8 tosses = 2^8 (since each toss has 2 possible outcomes: heads or tails) - 9 = 256 - 9 = 247

Step 2: Calculate the number of outcomes where exactly 2 of those tosses result in heads.
To find this, we need to determine the number of ways to choose 2 out of the 8 tosses that result in heads (i.e., C(8, 2)).

Number of outcomes with exactly 2 heads = C(8, 2) = 28

Step 3: Calculate the probability.
Now that we know the number of outcomes with exactly 2 heads and the total number of outcomes where there are at least 2 heads, we can calculate the probability:

Probability of getting exactly 2 heads given that there were at least 2 heads = (Number of outcomes with exactly 2 heads) / (Total number of outcomes where there are at least 2 heads)
= 28 / 247 ≈ 0.1133

Therefore, the probability of getting exactly 2 heads given that there were at least 2 heads in the 8 tosses is approximately 0.1133, or 11.33%.