Problem:

Find the minimum number of times a fair coin needs to be tossed so that the probability of getting at least two heads is at least 0.96.

Solution:

To solve this problem, we need to understand the concept of probability and the rules of tossing a fair coin.

Understanding Probability:

Probability is a measure of the likelihood that a specific event will occur. It is represented as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

Rules of Tossing a Fair Coin:

When a fair coin is tossed, there are two possible outcomes: heads (H) or tails (T). Each outcome has an equal probability of 0.5.

Calculating the Probability:

To calculate the probability of getting at least two heads, we need to consider the possible outcomes. Let's calculate the probability for each possible number of coin tosses:

1. One toss:
- There are two possible outcomes: H or T.
- The probability of getting at least two heads is 0 (impossible).

2. Two tosses:
- There are four possible outcomes: HH, HT, TH, or TT.
- The probability of getting at least two heads is 0.25 (HH).

3. Three tosses:
- There are eight possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT.
- The probability of getting at least two heads is 0.5 (HHH, HHT, HTH, or THH).

4. Four tosses:
- There are sixteen possible outcomes.
- The probability of getting at least two heads is 0.625 (HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, THHH, THTH, TTHH).

By calculating the probabilities for different numbers of coin tosses, we can see that the probability increases as the number of tosses increases.

Minimum Number of Tosses:

To find the minimum number of tosses required to achieve a probability of at least 0.96, we continue calculating the probabilities until we reach a value greater than or equal to 0.96.

5. Five tosses:
- There are thirty-two possible outcomes.
- The probability of getting at least two heads is 0.8125.

6. Six tosses:
- There are sixty-four possible outcomes.
- The probability of getting at least two heads is 0.9375.

7. Seven tosses:
- There are one hundred and twenty-eight possible outcomes.
- The probability of getting at least two heads is 0.984375.

8. Eight tosses:
- There are two hundred and fifty-six possible outcomes.
- The probability of getting at least two heads is 0.99609375.

Hence, by tossing the fair coin at least eight times, the probability of getting at least two heads is at least 0.96, as required.

Conclusion:

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is eight.