Question

A fair coin is tossed 27 times. In how many outcomes do at most 25 heads occur? a) 351 b) 379 c) 134,217,700 d) 134,217,349e) 28

Answer 1

When tossing a fair coin 27 times, the total number of outcomes with at most 25 heads is 134,217,700.

Step-by-step explanation:

When tossing a fair coin 27 times, we need to find the number of outcomes where at most 25 heads occur.

To solve this, we can use the concept of combinations. The total number of outcomes when tossing a fair coin 27 times is 2^27, since each toss has two possible outcomes (heads or tails).

So, we need to find the sum of the number of outcomes with 0, 1, 2, ..., 25 heads. This can be calculated using the formula for combinations:

nCr = n! / (r!(n-r)!)

The sum of these values will give us the total number of outcomes with at most 25 heads.

Calculating this can be time-consuming, but luckily there is a shortcut using Pascal's triangle. Pascal's triangle gives us the coefficients for each row of combinations.

The 27th row of Pascal's triangle is: 1 27 351 2925 17550 80730 ...

The sum of the first 26 coefficients (including 1) gives us the total number of outcomes with at most 25 heads, which is 134,217,700.

Answer 2

To find the number of outcomes with at most 25 heads when a fair coin is tossed 27 times, we can use the concept of combinations. By summing the number of outcomes with 0 to 25 heads, we find that there are d) 134,217,349 possible outcomes.

Step-by-step explanation:

When a fair coin is tossed, there are two possible outcomes: heads (H) or tails (T). The number of outcomes in which at most 25 heads occur when the coin is tossed 27 times can be determined using the concept of combinations.

To find the number of outcomes with at most 25 heads, we need to sum the number of outcomes with 0, 1, 2, ..., 25 heads. Using the formula for combinations, this can be calculated as:

C(27, 0) + C(27, 1) + C(27, 2) + ... + C(27, 25).

Using the formula for combinations, which is C(n, r) = n! / (r!(n-r)!), we can calculate the number of outcomes as:

C(27, 0) + C(27, 1) + C(27, 2) + ... + C(27, 25) = 1 + 27 + 351 + ... + 351 = 134,217,349.

Therefore, the correct answer is d) 134,217,349.