Question 1:
Binomial distribution
A Swiss mathematician Jakob Bernoulli had derived the formula of the Binomial
Distribution.
In an experiment or test that is repeated several times, a binomial distribution can be
viewed as simply the possibility of a SUCCESS or FAILURE outcome.
The binomial is a form of distribution with two possible results.
Formula := b(x; n, P) = nCx * Px * (1 – P)n – x
Where n is the total number of trials of the event. P is the probability for success in
individual trial and the x is the number of successes that result from the binomial
experiment.
Conditions to apply Binomial Distribution:-
The number of observations n is fixed.
Each observation is independent.
Each observation represents one of two outcomes (“success” or “failure”).
The probability of “success” p is the same for each outcome.
Examples:
The number of times the lights are green in 10 sets of traffic lights,
Number of teachers with uniforms in the staffroom of 30,
A number of plants with diseased leaves from a sample of 60 plants.
poisson distribution
This distribution was invented by the famous French mathematician Simon Denis Poissn.
This is often referred to as the distribution of rare events.
A Poisson cycle wherein continues but a finite interval of time or space, discrete events
occur. A Poisson is now recognized as a vitally important distribution in its own right.
Letting p represents the likelihood of winning at given any attempt, the mean or the
average number of wins in n tries will be given by π=np.
Using the binomial distribution of The Swiss mathematician Bernoulli, Poisson showed
that the chance of winning k is about,
Formula:= p(k)=e–π.πk/k!
Where e is the exponential function and k!=k(k-1)(k-2)…..1
Condition to apply Poisson Distribution:-
1. The probability of more than one occurrence in the small interval is negligible (i.e.
they are rare events).
2. Every event must be at random and independent from others.
3. The probability of the event taking place is proportional to the size of the interval
for a small interval.
4. Events are often failures, injuries, or extreme natural occurrences, such as
Earthquakes where there is no theoretical upper limit of the number of incidents.
Examples:-
1. The average number of lions seen on a 1-day safari is 5. What is the probability
that tourists will see fewer than four lions on the next 1-day safari?
2. The average number of computers sold by the Infotech Company is 15 homes per
day. What is the probability that exactly 20 homes will be sold tomorrow?
Relationship between Binomial distribution and Poisson distribution:
The binomial distribution tends toward the Poisson distribution as n → ∞, p → 0 and np
stays constant.
If n is large and p is small, the Poisson distribution with λ= np closely approximates the
binomial distribution.
Usually, the Poisson distribution is used to estimate the true underlying truth.
Determining whether a random variable has a Poisson distribution can be difficult.
Question 2 :
λ = 200
t = 0.01
λt = 2
P(X = x) = e-λt (λt)x
X!
(a) Probability that exactly three will not show up
P(x=3) => e-2 (2)3
3!
= 0.18044 ANS
(b) Probability that fewer than 4 will show up :
P(x<4) = P(x=0) + P(x=1) + P(x=2) + P(x=3)
= e-2 [ 20/0! + 21/1! + 22/2! + 23/3!]
= 1.0375 ANS
C) Probability that at least four will show up :
P(x ≥ 4) = 1 - P(x=0) + P(x=1) + P(x=2) + P(x=3)
= 1 - e-2 [ 20/0! + 21/1! + 22/2! + 23/3!]
= 1 – 1.0375
= - 0.0375 ANS
Question 3 :
1. Probability that all men will be alive:
P=(2/3)^5 = 32/243.
2. Probability that at least 3 men will be alive:
P(x>=3) = C(5,3) * (2/3)^3 * (1/3)^2 + C(5,4) * (2/3)^4 * (1/3)^1 + C(5,5) *
(2/3)^5
= 80/243+80/243+32/243
=192/243.
3. Only two men will be alive :
P(x=2)=C(5,2) * (2/3)^2 * (1/3)^3
= 40/243.
4. At least 1 man will be alive:
P(x>=1) = 1 - P(x=0)
= 1 - C(5,0) * (1/3)^5
= 242/243.
