6: Binomial & Poisson Distribution

Question 1

What is the formula for the probability of a successful outcome?

Answer 1

P (A) = M / N

Probability of event ‘A = Number of successful outcomes / Number of total outcomes

Question 2

What are mutually exclusive and non-mutually exclusive events?

Answer 2

Mutually exclusive: Outcomes cannot occur at the same time

Non mutually exclusive: Outcomes can occur at the same time

Question 3

What type of connective do mutually exclusive and non mutually exclusive events apply to?

Answer 3

‘or’

Question 4

What is the formula for mutually exclusive events using the ‘or’ connective?

Answer 4

P (A or B) = P(A) + P(B)

Question 5

What is the formula for NON mutually exclusive events using the ‘or’ connective?

Answer 5

P (A or B) = P(A) + P(B) - P(A and B)

Question 6

What are independent and dependent events?

Answer 6

Independent: The outcome of one event does not effect the probability of another occurring

Dependent: Occurrence of one event effects the probability of another occurring

Question 7

What is the formula for an independent event using the ‘and’ connective?

Answer 7

P (A and B) = P(A) * P(B)

Question 8

What is the formula for a dependent event using the ‘and’ connective?

Answer 8

P (A and B) = P(A) * P(B/A)

P(B/A) = probability of B given A has occured.

Question 9

What is the probability of an event, A, not occurring?

Answer 9

P(not A) = 1 - P(A)

Question 10

Give an example for each of the following:
1) Mutually exclusive event (or)
2) Non mutually exclusive event (or)
3) Independent event (and)
4) Dependent event (and)

Answer 10

1) Tossing a coin
2) Drawing 1 card based on colour & suite
3) Rolling 2 dice
4) Drawing cards without replacement

Question 11

What is the formula for the binomial coefficient?

What does each letter stand for?

Answer 11

^nCr = n! / (n-r)!*r

r = specified successes
n = number of independent identical trials

Question 12

In binomial distributions, what is the formula for the probability of having exactly ‘r’ successes in a given number of trials?

Answer 12

P(r) = ^nCr * p^r * q^n-r

Where:
p = probability of success in a single trial
&
q = probability of failure in a single trial

Question 13

What is the formula for ‘q’, the probability of failure in a single trial?

Answer 13

q = 1 - p

Question 14

How would you calculate the probability of the following events in a sample of 6:
1) No successful events
2) 1 successful event
3) More than 4 successful events
4) At least 2 successful events (= 2 or more)

Answer 14

1) P(0)
2) P(1)
3) P(5) + P(6)
4) 1 - P(0) + P(1)

Question 15

When should the binomial distribution be approximated by the Poisson probability distribution?

Answer 15

When n*p < 5

Question 16

What is the formula for the Poisson probability distribution?

What is Lambda equal to?

Answer 16

P(r) = (Lambda^r * e^-Lambda) / r!

Lamda is equal to the mean value, usually n*p

Question 17

What is the first thing you should do in a Poisson probability question?

Answer 17

Check that Lambda is less than 5

Question 18

How is lambda calculated in questions where probability of events is given during a fixed interval of time?

Use a call centre as an example with 30 calls in one hour and intervals of 1 minute and 5 minutes respectively.

Answer 18

Lambda = average rate of events

30 calls/60 mins = 0.5 call/min

for 1 min interval Lambda = 1*0.5=0.5 call/min

for 5 min Lambda = 5*0.5=2.5 call/5 min

Question 19

How would you calculate the probability of the following events:
1) Probability of having 2 or 3 calls in a minute
2) Probability of having at least one call in 5 minutes

Answer 19

1) P (2 or 3) = P(2) + P(3)

2) P(at least 1) = 1 - P(no calls)

Question 20

Given two companies with varying accident figures over the same time period, how would you calculate the associated success rates and probabilities for each?

Answer 20

Calculate the success rate: P(A) = M/N for each company using their respective information and note them down as P(A) and P(B) (for example)

Use the same probability formula for each, based on Binomial or Poisson’s distribution, but remember to use the ‘n’ and ‘r’ values for each respective data set.

Question 21

How would you calculate the probability of both Company X AND Company Y expecting no accidents?

Answer 21

Company X P(0) * Company Y P(0)

Question 22

How would you calculate the probability of a shipment containing less than 2 broken lamps on arrival?

Answer 22

P(0) + P(1)

LESS THAN 2, meaning only 0 or 1