Consider a sequence of eight independent coin tosses. What is the probability we get the sequence TTTTTTTT?

Question

Consider a sequence of eight independent coin tosses.  What is the probability we get the sequence TTTTTTTT?

BlackTom AI · Accepted Answer

Each independent fair coin toss has probability 1/2 of landing tails, so the probability of eight tails in a row is the product of eight 1/2 factors: (1/2)^8.

1) Relevant knowledge points and concepts
- Independence: coin tosses are independent, so probabilities multiply for a specific sequence.
- Multiplication rule for independent events: P(A and B and ...) = P(A) × P(B) × ...
- Powers of 2: (1/2)^n = 1/2^n.

2) Complete step-by-step reasoning
- For one toss: P(T) = 1/2.
- For two independent tosses: P(TT) = (1/2) × (1/2) = (1/2)^2.
- Extend to eight tosses: P(TTTTTTTT) = (1/2)^8.

3) All intermediate calculation steps
- (1/2)^8 = 1 / (2^8).
- 2^8 = 256, so (1/2)^8 = 1/256.
- Compute decimal: 1 ÷ 256 = 0.00390625.
- Rounded to four decimal places this is 0.0039.

4) Why the correct answer is correct
Because each toss independently has probability 1/2 of tails and we require eight specific outcomes in sequence, multiplication of eight 1/2 factors yields 1/256 ≈ 0.00390625, which matches the listed option 0.0039 when rounded.

5) Why the other options are incorrect
- 0.5: This is the probability of a single toss being tails, not eight in a row.
- 0.03125: This equals 1/32 = (1/2)^5, which would be the probability of five tails in a row, not eight.
- 0: This would mean the event is impossible; getting eight tails in a row is possible (just unlikely), so probability is positive (1/256).

6) Conclusion
The probability of the sequence TTTTTTTT in eight independent fair coin tosses is 1/256 ≈ 0.00390625, rounded to 0.0039. This confirms the provided answer 0.0039.

Consider a sequence of eight independent coin tosses. What is the probability we get the sequence TTTTTTTT?单项选择题

类似问题

Grades on a very large statistics course have historically been awarded according to the following distribution. HD D C P Z or Fail 0.15 0.20 0.30 0.30 0.05 What is the probability that a student scores higher than a Credit (C)?

Question at position 3 Suppose is a random variable representing rolling a six-sided dice. Assume the dice is fair, so there is an equal probability of rolling any number 1 through 6. What is the following probability? Clear my selection

The following table contains the probability distribution for X = the number data packages transmitted through satellite What is the probability (between 0 and 1 with 2 decimal places) of at least one successful transmission? X 0 1 2 3 p(X) 0.35 0.35 0.25 0.05

The following data are based on a survey taken by a consumer research firm. X = number of internet connected devices in a household and P(X) = probability of U.S. households. What value for P(X = 2 ) will make this a PDF? X 0 1 2 3 4 5 P(X) 0.03 0.11 ? 0.39 0.12 0.07

Let Y have the PMF 𝑝 𝑌 ( 𝑦 ) = 𝑃 ( 𝑌 = 𝑦 ) = 𝑦 10 for y = 1, 2, 3, 4. What is 𝑃 ( 𝑌 ≤ 2 ) ?

Suppose we roll a six-sided die twice, each time recording the value that lands face up. Let X be the sum of the two rolls. What is the support of X?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单