Question

a) Recent research suggests that the global monthly average number of airplane crashes of commercial airliners is 3.5.

i) State the random variable and its probability distribution function, with suitable parameters. [2 marks]

What is the probability that there will be: ii) at least 2 such accidents in the next month? [3 marks]

iii) at most 1 accident in the next month? [3 marks]

b) A certain state in the United States of America reported that approximately 80,000 marriages occurred in the last year. Find the approximate probability that, for at least one of the couples, both partners were born on December 25.

Answer #1

i) here as average number of accident is constant and independent from year to year

this follows poisson distribution with paramter =3.5

ii)

P( at least 2 such accidents in the next month)=P(X>=2)=1-P(X<=1)=1-(P(X=0)+P(X=1))

=1-(e^{-3.5}3.5^{0}/0!+e^{-3.5}3.5^{1}/1!)=0.864112

iii)P( at most 1 accident in the next month) =1-P(at least 2) =1-0.864112 =0.135888

b)

here expected number of couples that born on December 25
=80000/(365)^{2} =0.6005

hence probability that, for at least one of the couples, both partners were born on December 25

=1-P(none born on December 25 )
=1-e^{-0.6005}*0.6005^{0}/0!
=0.4515

Assume that the monthly worldwide average number of airplane
crashes of commercial airlines is 2.2. What is the probability that
there will be
(a) at least 3 such accidents in the next month?
(b) at most 5 such accidents in the next 2 months?
(c) exactly 7 such accidents in the next 4 months?

Assume that the monthly worldwide average number of airplane
crashes of commercial airlines is 2.2. What is the probability that
there will be
(a) more than 3 such accidents in the next month? 0.1807
(b) less than 2 such accidents in the next 3 months?
(c) exactly 6 such accidents in the next 4 months?

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