Consider the following scenarios. For each scenario, decide if it follows a bino

Question

Consider the following scenarios. For each scenario, decide if it follows a binomial or a Poisson distri- bution. Also calculate the corresponding probabilities.

(a) A bag contains 6 red and 3 black counters. I take a counter, note its color and return it, and repeat this until I have selected 5 counters. Find the probability that I have selected one red counter.

(b) While crossing a bridge the probability that a car will get a puncture is 0.00005. During a given day, 10,000 cars cross the bridge. Find the probability that at least one car will get a puncture.

(c) On average a tool hire firm receives requests to hire out 2 floor sanders a day. Find the probability that on a given day there are at most 2 requests for sanders.

(d) A bad stretch of road has, on average, 65 accidents per year. Find the probability that during a given week there will be no accidents. You may assume there are 52 weeks in a year.

Explanation / Answer

Ans:

a)Binomial distribution

p=6/9=2/3

1-p=1/3

P(X=1)=5C1*(2/3)1*(1/3)4

=0.0412

b)Binomial distribution

n=10000,p=0.00005

P(atleast one car gets puncture)=1-P(none puncture)

=1-10000C0*(0.00005)0*(1-0.00005)10000

=1-0.9999510000

=1-0.6065

=0.3935

c)Possion distribution

mean=2 floor sanders per day

P(k<=2)=e-2*[20/0!+21/1!+22/2!]=e-2*[1+2+2]=e-2*5=0.6767

or using excel function:

POISSON(2,2,TRUE)=0.6767

d)Poission distribution:

mean=65 accidents per year=65/52=1.25 accidents per week

P(k=0)=e-1.25 *(1.25)0/0!

=e-1.25

=0.2865

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