The probability of getting a “head” in a single toss of a biased coin is 0.3. The coin is tossed repeatedly till a “head” is obtained. If the tosses are independent, then the probability of getting "head" for the first time in the fifth toss is ___________
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.
Important Questions from Basics of Probability
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ______ . (rounded off to three decimal places)
- A quadratic polynomial $(x - \alpha)(x - \beta)$ over complex numbers is said to be square invariant if $(x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.
The probability that the roots of the chosen polynomial are equal is ____________ (rounded off to one decimal place) - If, out of 50 fish of a pond, 12 had no ectoparasite and remaining had varying numbers of parasites on them, then how many of fish had at least one parasite ?
For a distribution with mean, median, mode and standard deviation 25, 24, 26 and 5 respectively, Karl Pearson's coefficient of skewness equals to:
Suppose two distributions have the same mean, same standard deviation and are equally skewed, but the first distribution is more peaked than the
other. This feature of the frequency distributions in which they are different are called:
