Question

Please give help for this question.

Question 4. Coin tossing, again. In class on Monday, January 29th, we discussed an example showing that the conditional independence of events does not imply their unconditional independence. As a reminder, the setup of the example was as follows. We had two coins, coin A and coin B. We chose a coin at random (i.e., with probability 0.5) and tossed the chosen coin repeatedly. Given the choice of a coin, the coin tosses were independent, with the following probabilities: P(H | if coin A is chosen) 0.9. P(H | if coin B is chosen) = 0.1. The setup of this experiment is illustrated by the following figure (same as figure 8 on page 10 of your reading assignment on independence) 0.9 0.9 Coin A 0.9 0.5 0.1 0.5 0.1 0.9 Coin B 0.9 0.9 We computed the probability of the event "toss 11 is H" by conditioning this event on the choice of the coin and applying the total probability the- orem: P( toss 11 is H) P(A)P( toss 11 is H | A) P(B) P( toss 11 is H | B) 0.5 × 0.9 +0.5 × 0.1 = 0.5. Suppose you walk into a room where this experiment is being conducted. You walked in the moment after the coin had been chosen but before the tosses had begun. You have no idea which coin was chosen but you ob- serve 10 Hs in a row. Compute the following probability exactly: P( toss 11 is H 1 first 10 tosses are Hs) =

Answer 1

ara Head Ay (o. s) (o-l

︵ o.