Heads in Repeated Tosses

Solution & Explanation

To solve the problem of finding the likelihood of getting heads exactly 5 times out of 6 tosses with a biased coin that shows heads with a probability of 30%, we can use the binomial probability formula. This formula is applicable because:

1. Fixed Number of Trials (n): We are flipping the coin 6 times.
2. Two Possible Outcomes: Each flip results in either heads or tails.
3. Constant Probability of Success (p): The probability of getting heads (success) is constant at 0.3 for each flip.
4. Independent Trials: Each coin flip is independent of others.

Binomial Probability Formula

The probability of getting exactly k successes (heads) in n trials (flips) is given by the formula:

$P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$

Where:

• $n$ = number of trials (6)
• $k$ = number of successes (5)
• $p$ = probability of success on a single trial (0.3)
• $\binom{n}{k}$ = $n$ choose $k$ = $\frac{n!}{k!(n-k)!}$

Calculating the Probability

1. Calculate $\binom{n}{k}$:

   • $\binom{6}{5} = \frac{6!}{5!1!} = 6$

2. Calculate $p^k$:

   • $0.3^5 = 0.00243$

3. Calculate $(1-p)^{n-k}$:

   • $(1-0.3)^{6-5} = 0.7^1 = 0.7$

4. Combine the Components:

   • $P(X = 5) = 6 \cdot 0.00243 \cdot 0.7$
   • $P(X = 5) = 6 \cdot 0.001701$
   • $P(X = 5) = 0.010206$

Conclusion

The probability of getting exactly 5 heads in 6 tosses of a biased coin with a 30% chance of landing on heads is approximately 1.02%. This low probability reflects the difficulty of achieving such an outcome given the bias of the coin towards tails.