Animated Solution for Mathematics - Probability: Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is :

To determine how many times a coin must be tossed to be more than $99\%$ sure of getting at least one head, we avoid the tedious process of calculating individual probabilities for one, two, or three heads.

Instead, we utilize the Complement Rule. We recognize that the probability of getting at least one head is the complement of getting zero heads (all tails).

The fundamental relationship is defined as:

For a fair coin, the probability of landing on tails in a single toss is $\frac{1}{2}$. For $n$ independent tosses, the probability of obtaining all tails is given by:

We require this probability to satisfy the condition of being greater than $0.99$:

Subtracting $1$ from both sides of the inequality yields:

When multiplying both sides by $-1$, we must remember to flip the inequality sign. This results in:

Converting the decimal $0.01$ to the fraction $\frac{1}{100}$, we have:

Taking the reciprocal of both sides flips the inequality sign once more, leading to:

We now seek the smallest integer $n$ that satisfies the inequality $2^n>100$. Evaluating the powers of $2$:

Since $128>100$, the smallest integer $n$ that satisfies our condition is $7$.