A bag contains 5 coins, of which 4 are fair; the remaining coin has a head on both sides. If a coin is selected at random from the bag and tossed three times, construct a probability tree to determine the probability that heads will occur at least twice.
A) Pr(at least two heads) =
B) Pr(at least two heads) =
C) Pr(at least two heads) =
D) Pr(at least two heads) =
E) Pr(at least two heads) =

Question

A bag contains 5 coins, of which 4 are fair; the remaining coin has a head on both sides. If a coin is selected at random from the bag and tossed three times, construct a probability tree to determine the probability that heads will occur at least twice. ​
A) Pr(at least two heads) = 
B) Pr(at least two heads) = 
C) Pr(at least two heads) = 
D) Pr(at least two heads) = 
E) Pr(at least two heads) =

Quizplus · Accepted Answer

First, we need to determine the probability of selecting each coin from the bag:

- Probability of selecting a fair coin = 4/5 
- Probability of selecting the double-headed coin = 1/5

Using this information, we can construct the probability tree:

[Please refer to the image in the link below]

The branches represent the outcomes of each coin toss, and the probabilities are calculated by multiplying the probabilities of the previous event and the corresponding outcome.

To find the probability of getting at least two heads, we need to add up the probabilities of the following outcomes:

- HHH (three heads) 
- HHT (two heads, one tail) 
- HTH (two heads, one tail) 
- THH (two heads, one tail)

The probability of getting three heads when tossing a fair coin is (1/2)^3 = 1/8. The probability of getting two heads and one tail when tossing a fair coin is 3*(1/2)^3 = 3/8 (since there are three ways to arrange two heads and one tail). The probability of getting two heads when tossing the double-headed coin is 1.

Using these probabilities, we can calculate the overall probability of getting at least two heads:

- Probability of getting at least two heads = (4/5)*(3/8) + (4/5)*(1/8) + (1/5)*1 
- Probability of getting at least two heads = 3/10 + 1/5 + 1/5 
- Probability of getting at least two heads = 11/20

Therefore, the correct answer is C.

A Bag Contains 5 Coins, of Which 4 Are Fair;