Assume You Toss A Fair Coin 5 Times, 3 Points

Assume That You Toss A Fair Coin 5 Timesa 3 Pts W

Assume that you toss a fair coin 5 times. (a) What is the probability that you get 5 heads? (Show work and write the answer in simplest fraction form). (b) What is the probability of getting heads in the 5th toss, given that the first four tosses are tails? (Show work and write the answer in simplest fraction form). (c) If event A is “Getting heads in the 5th toss” and event B is “The first four tosses are tails”, are event A and event B independent? Please explain.

Paper For Above instruction

The problem involves analyzing probabilities related to tosses of a fair coin. It encompasses basic probability calculations, conditional probability, and the concept of independence.

For part (a), calculating the probability of getting 5 heads in 5 coin tosses involves recognizing that each toss is independent and has a probability of 1/2 for heads. The probability of all five occurring as heads is (1/2)^5 = 1/32. This is because the outcomes of each independent toss multiply, leading to the fraction 1/32.

Part (b) asks for the probability that the fifth toss is heads, given that the first four are tails. Since coin tosses are independent events, the outcome of the fifth toss does not depend on previous outcomes. Therefore, the probability remains 1/2. The conditional probability P(Heads in 5th | first four are tails) equals P(Heads in 5th) = 1/2.

For part (c), the question explores whether events A ("Getting heads in the 5th toss") and B ("First four are tails") are independent. Two events are independent if the occurrence of one does not affect the probability of the other. Calculate P(A): the probability of heads in the fifth toss, which is 1/2; and P(B): the probability that the first four are tails, which is (1/2)^4 = 1/16. Next, find P(A ∩ B): the probability that the first four are tails and the fifth is heads, which is (1/2)^4 (1/2) = (1/2)^5 = 1/32. Since P(A ∩ B) = P(A) P(B) = (1/2) * (1/16) = 1/32, the equation holds, confirming that events A and B are independent. The independence confirms that the outcome of the fifth toss does not depend on the previous four, consistent with properties of a fair coin.

The understanding of these basic probability principles forms the foundation of more complex probabilistic analyses and is essential for modeling real-world scenarios involving random binary events.

Probability Calculations and Analysis

In the context of coin tosses, the probability of specific outcomes exemplifies fundamental principles of probability theory. The probability of getting five heads in five tosses exemplifies the multiplicative rule for independent events, highlighting how probabilities compound over multiple trials. This probability, 1/32, signifies that such an outcome is relatively rare but possible with a fair coin, which has unbiased 50% heads and tails. These calculations form the basis of understanding combinatorial probability in simple binary experiments.

Conditional probability offers insight into how past outcomes influence (or do not influence) future results. Since coin tosses are independent, knowledge that the first four are tails does not alter the probability of the fifth being heads, reaffirming the independence inherent in Bernoulli trials. This principle is critical in areas like gambling, statistical inference, and modeling of independent random processes.

Furthermore, the concept of independence was verified through joint and marginal probabilities, illustrating core statistical properties. The key takeaway is that for a fair coin, each toss remains unaffected by previous tosses, with probabilities that remain constant at 1/2 for each individual event.

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