Understanding the Problem

The question asks for the probability of a specific sequence of events when tossing a biased coin multiple times. The coin has a known probability of landing heads, and the goal is to find the chance of achieving the first head only on the fifth toss.

Identifying Key Information

• Probability of getting a "head" (P(H)) = 0.3
• Probability of getting a "tail" (P(T)) = 1 - P(H) = 1 - 0.3 = 0.7
• The coin tosses are independent events.
• We need the probability of the first head occurring on the fifth toss.

Calculating the Probability

For the first head to appear on the fifth toss, the sequence must be: Tail, Tail, Tail, Tail, Head (T, T, T, T, H).

Since the tosses are independent, we multiply the probabilities of each individual outcome:

Probability = P(T) $\times$ P(T) $\times$ P(T) $\times$ P(T) $\times$ P(H)

Probability = $(P(T))^4 \times P(H)$

Substitute the values:

Probability = $(0.7)^4 \times (0.3)$

Calculate $(0.7)^4$:

$(0.7)^4 = 0.7 \times 0.7 \times 0.7 \times 0.7 = 0.2401$

Now, multiply by P(H):

Probability = $0.2401 \times 0.3 = 0.07203$

Conclusion

The calculated probability is 0.07203. This value falls between 0.07 and 0.08, matching the provided answer range.