Question

A piñata is a container filled with toys and candy and is broken open by hitting it with a stick. Sophia is trying to break the piñata. The probability that she will break the piñata on the first hit is 0.7. She will continue to hit the piñata until it breaks. If she does not break the piñata on a particular hit, the piñata is weakened and the probability that she will break it on the next hit is 0.1 greater than the probability on the previous hit. For example, if the piñata does not break on the first hit, the probability that it will break on the second hit is 0.8.
(a) Calculate the probability that Sophia docs.not break the piñata on the first hit and does break the piñata on the second hit. Let the random variable X represent the number of hits required for Sophia to break the piñata.
(b) Complete the probability distribution of X in the table below Probability of x 0.7

Answer 1

Answer:

A) 0.24

B) The probability mass function of X, the number of hits required to break the piñata

X | P(X)

0 | 0.00

1 | 0.70

2 | 0.24

3 | 0.054

4 | 0.006

Step-by-step explanation:

Probability of Sophia breaking the piñata on the first attempt = 0.7

Probability of Sophia NOT breaking the piñata on the first attempt = 1 - 0.7 = 0.3

Probability of Sophia breaking the piñata on the second atrempt = 0.8

Probability of Sophia NOT breaking the piñata on the second atrempt = 1 - 0.8 = 0.2

Probability of Sophia breaking the piñata on the third atrempt = 0.9

Probability of Sophia NOT breaking the piñata on the second atrempt = 1 - 0.9 = 0.1

Probability of Sophia breaking the piñata on the fourth attempt = 1.0 (this is the highest number of attempts as a probability 1.0 means that the piñata breaks on the fourth attempt if it hasn't broken by now)

A) The probability that Sophia does not break the piñata on the first hit and does break the piñata on the second hit.

The required probability = (Probability that Sophia does not break the piñata on the first hit) × (Probability that Sophia does break the piñata on the second hit)

= 0.3 × 0.8 = 0.24

B) Let the random variable X represent the number of hits required for Sophia to break the piñata. Complete the probability distribution of X in the table below Probability of x 0.7

- X = 0, P(X) = 0

- X = 1

Probability of Sophia breaking the piñata on the first hit = 0.7

- X = 2

Probability of Sophia not breaking the piñata on the first hit, but breaking it on the second hit = 0.3 × 0.8 = 0.24

- X = 3

Probability of Sophia not breaking the piñata on the first and second hit, but breaking it on the third hit = 0.3 × 0.2 × 0.9 = 0.054

- X = 4

Probability of Sophia not breaking the piñata on the first, second and third hit, but breaking it on the fourth hit = 0.3 × 0.2 × 0.1 × 1.00 = 0.006

The probability mass function is then

X | P(X)

0 | 0.00

1 | 0.70

2 | 0.24

3 | 0.054

4 | 0.006

To check of we are correct, the probabilities should sum up to give 1.0

The cumulative probability

= 0.00 + 0.70 + 0.24 + 0.054 + 0.006 = 1.00

Hope this Helps!!!!