Shreya Khamrai

2022-07-03

What is the probability that there will be strictly more heads than tails out of 10 flips
of a fair coin? Out of 20 flips?

Vasquez

To calculate the probability of getting strictly more heads than tails out of a given number of coin flips, we need to consider the different possible outcomes and determine the favorable outcomes.
Let's first consider 10 flips of a fair coin. Each flip has two possible outcomes: heads (H) or tails (T). Therefore, the total number of possible outcomes for 10 flips is ${2}^{10}$.
To find the probability of getting strictly more heads than tails, we need to count the favorable outcomes where the number of heads is greater than the number of tails. We can approach this by counting the favorable outcomes for each possible number of heads (from 6 to 10) and summing them up.
For 10 flips, the favorable outcomes are:
- 6 heads: We can choose 6 out of 10 flips to be heads, which can be calculated as $\left(\genfrac{}{}{0}{}{10}{6}\right)$.
- 7 heads: We can choose 7 out of 10 flips to be heads, which can be calculated as $\left(\genfrac{}{}{0}{}{10}{7}\right)$.
- 8 heads: We can choose 8 out of 10 flips to be heads, which can be calculated as $\left(\genfrac{}{}{0}{}{10}{8}\right)$.
- 9 heads: We can choose 9 out of 10 flips to be heads, which can be calculated as $\left(\genfrac{}{}{0}{}{10}{9}\right)$.
- 10 heads: All 10 flips are heads, which is only one outcome.
Therefore, the total number of favorable outcomes for 10 flips is $\left(\genfrac{}{}{0}{}{10}{6}\right)+\left(\genfrac{}{}{0}{}{10}{7}\right)+\left(\genfrac{}{}{0}{}{10}{8}\right)+\left(\genfrac{}{}{0}{}{10}{9}\right)+\left(\genfrac{}{}{0}{}{10}{10}\right)$.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:

$\text{Probability}=\frac{\left(\genfrac{}{}{0}{}{10}{6}\right)+\left(\genfrac{}{}{0}{}{10}{7}\right)+\left(\genfrac{}{}{0}{}{10}{8}\right)+\left(\genfrac{}{}{0}{}{10}{9}\right)+\left(\genfrac{}{}{0}{}{10}{10}\right)}{{2}^{10}}$
Similarly, we can calculate the probability for 20 flips by considering the favorable outcomes for each possible number of heads from 11 to 20.

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