Question

You’re a magician trying to figure out one of your competitor’s most dazzling tricks. As a last ditcheffort, you’ve made a shady deal with someone who looks suspiciously like Tucker in a back alley tosteal your competitor’s prop. He gives you a bag containing 6 regular coins and 4 two-faced coins (bothsides are heads), leaving you to figure out the trick for yourself.

(a) Suppose you choose a random coin from the bag and flip it. What is the probability of getting aheads?

(b) Suppose you choose a random coin from the bag. Then, you flip this coin twice and both resultsare heads. What is the probability that the coin is two-faced?

(c) Suppose you flip a particular coinntimes and all of the results are heads. How many flips do youneed to see come up heads before the probability that the coin is two-faced is greater than 90%?

Answer 1

The bag contains 6 regular coins and 4 two-faced coins. There are a total a 10 coins (6+4=10) in the bag.

The probability that a randomly chosen coin from the bag is a regular coin is

The probability that a randomly chosen coin from the bag is a 2-faced coin is

The conditional probability of getting a head given that the coin is regular is

The conditional probability of getting a head given that the coin is 2-faced is (both sides are heads)

(a) Suppose you choose a random coin from the bag and flip it. What is the probability of getting a heads?

The probability of getting a heads is

ans: the probability of getting a heads is 0.7

(b) Suppose you choose a random coin from the bag. Then, you flip this coin twice and both results are heads. What is the probability that the coin is two-faced?

Let the event of getting heads in both the tosses be 2-heads

the probability that the coin is two-faced given that the result is 2-heads is

ans: the probability that the coin is two-faced given that both results are heads is 0.7273

(c) Suppose you flip a particular coin n times and all of the results are heads. How many flips do you need to see come up heads before the probability that the coin is two-faced is greater than 90%?

Let the event of getting n heads in n tosses be n-heads.

We need the probability that the coin is two-faced given that n-heads is greater than 90%

the probability that the coin is two-faced given that n-heads is

We want this probability to be greater than 0.90

ans: We need to see 4 flips come up heads before the probability that the coin is two-faced is greater than 90%.