Question

Losing a coin. You begin the day carrying 2 nickels and 1 dime in your pocket. Sadly, your pocket has a hole. Very sadly, at the end of the day, one of the 3 coins has fallen out. Each of the three coins has an equal probability of falling out. In the evening, you reach into your pocket and pull out one of the two remaining coins, each of which has an equal probability of being chosen. The coin you happen to pick from your pocket is a nickel.

Let D be the event you lost the dime; N be the event you lost a nickel; d be the event you pick a dime from your pocket, and n be the event you pick a nickel from your pocket.

A. Show a tree diagram for this scenario. The first stage shows which coin you lost, and the second stage shows which coin you pick from your pocket. Show the appropriate probability for each of the branches and for each of the outcomes. For the probabilities, you can use either algebraic symbols such as P(D), P(n|D), etc., or their numerical values, or both.

B. Given that you have picked a nickel from your pocket, what is the (conditional) probability that you LOST the DIME through the hold in your pocket. Show the equation (algebraic or numeric) that you use to calculate that result.   

Answer 1

Solution :

(a)

There are total 3 coins with 2 nickels and 1 dime. In first stage,

Probability to lose a nickel = 2/3

Probability to lose a dime = 1/3

In the first stage, if you lose dime, so there will be only nickels, so, the probability of picking a nickel in second stage is 1.

If you lose a nickel, there will be 2 coins (a nickel and a dime), each have equal probability to get picked up.

The tree diagram for this scenario is,

Thank You.