Question

In: Statistics and Probability

QUESTION THREE You are concerned about the possibility of experiencing on Coronavirus (COVID-19) during the next...

QUESTION THREE

  1. You are concerned about the possibility of experiencing on Coronavirus (COVID-19) during the next year. The probability of a COVID-19 outbreak is 0.5%. Your local insurer offers to pay you K300, 000 if a registered member of your family contracted COVID-19. The insurance policy costs K1, 500.00. If the inflation rate is forecast to be 10% during the next year, is the price of insurance policy fair.   
  2. A coin is biased so that it has a 60% chance of landing on heads. If it is thrown three times, find the probability of getting
  1. Three heads                                                                                            
  2. 2 heads and a tail                                                                                    

At least one head                                                                                    

Solutions

Expert Solution

Question (3A)

The probability of you experiencing COVID-19 is 0.5% which is 0.005

So the probability of you not experiencing COVID-19 = 1- 0.005 = 0.995

Now we calculate the expected value of the insurance for you

The value for the You column is (Benefits - Cost)

If you experience COVID-19, then the effective value for you = 300000 - 1500 = 298500

You are getting a beneft of 300000 and you are paying 1500, so the effective value for you = 300000 - 1500 = 298500

If you do not experience COVID-19, then the effective value for you = 0 - 1500 = -1500

You won't get anything if you don't experience COVID-19, so benefit is 0, cost is 1500

Case Probability P(x) Effective Value for you (V) P(x) * E
Experiencing COVID-19 0.005 298500 1492.5
Not Experiencing COVID-19 0.995 -1500 - 1492.5
P(x) * E = 0

The expected value from the insurance for you = P(x) * E = 0

So the insurance is fair since you are getting insured and you would be risk-free

Question (3B)

Given that a coin is biased so that it has a 60% chance of landing on heads

So P(H) = 0.6

The coin toss is a binomial distribution since there are only 2 outcomes possible here. Head and Tail

Probability of x successes in n trails for a binomial distribution is * px * qn-x

= n! / x! (n-x)!

Here p is probability of success and q is 1-p

Question (a)

Probability of three heads

Here p = 0.6 probability of a head

q = 0.4

n = 3 number of times a coin is tossed

x = 3 number of heads or number of successess

So Probability of three heads = * 0.63 * 0.43-3

= 3! / 3! (3-3)! * 0.216 * 1

= 0.216

So Probability of three heads = 0.216

Question (b)

Probability of two heads and a tail is same as calculating the probability of two heads since the third one would automatically be a trial

Here p = 0.6 probability of a head

q = 0.4

n = 3 number of times a coin is tossed

x = 3 number of heads or number of successess

So Probability of two heads = * 0.62 * 0.43-2

= 3! / 2! (3-2)! * 0.36 * 0.4

= 0.432

So Probability of two heads and one tail = 0.432

Question (c)

Probability of atleast 1 head is obtained by adding porbabilty of 1 head and 2 head and 3 heads

Here p = 0.6 probability of a head

q = 0.4

n = 3 number of times a coin is tossed

x = 3 number of heads or number of successess

So Probability of atleast 1 head = * 0.61 * 0.43-1  + * 0.62 * 0.43-2 + * 0.63 * 0.43-3

= 3! / 1! (3-1)! * 0.6 * 0.16 + 3! / 2! (3-2)! * 0.36 * 0.4 + 3! / 3! (3-3)! * 0.216 * 1

= 0.288 + 0.432 + 0.216

= 0.936

So Probability of atleast 1 head = 0.936


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