Q.3.89 - A First Course in Probability

Question

In Laplace’s rule of succession (Example 5e), are the outcomes of the successive flips independent? Explain.

Step-by-Step Solution

Answer

Successive coin tosses are not independent. But they are independent give knowledge which coin is chosen.

Step 1: Given Information

Are the outcomes of the successive flips independent? 

Step 2: Explanation

Laplace's rule of succession.

If two experiments are done, and the outcome of the second one is dependent on the outcome of the first one, the result in the second experiment is not independent.

In the example, we choose a coin, and each coin has a different probability of flipping heads.

The first experiment is choosing a coin and the second tossing it multiple times.

The flips in the second experiment are

Independent given the result of the first experiment, choosing the coin.

Step 3: Explanation

The probability of getting heads, in general, is the weighted average of probabilities of tossing heads on different coins, with weights being the probabilities that that coin is chosen - the Bayes formula.

$P\left(H\right)=\sum _{i=0}^{k}P(H\mid C_i)P\left(C_i\right)$

Given the result of some successive tosses -S, P(C_i)$\to$$P\left(C_i\right)\to P_S\left(C_i\right)=P\left(C_i/S\right)$ changes, the weights - probabilities that the i-th coin is chosen, change, and so does probability.

For example, if at least one head is tossed, the probability $P_{s(}{C}_i)=0$, therefore, $P_s\left(H\right)>P\left(H\right)$ (the other weights get larger too).

Step 4: Final Answer

Successive coin tosses are not independent. But they are independent give knowledge which coin is chosen.