Download Binomial Distribution Conditions of binomial distribution: There are

Binomial Distribution
Conditions of binomial distribution: There are four basic conditions for binomial distribution.
i. A finite number, 𝑛 , trials are carried out.
ii. The trials are independent.
iii. There must be exactly two outcomes of each trial. (Success and failure).
iv. The probability, 𝑝, of a successful outcome is the same for each trial.
The discrete random variable, 𝑋, is the number of successful outcomes in n trials. If above four
conditions are satisfied by the set 𝑋 then 𝑋 is said to follow the binomial distribution.
It is written as 𝑋~𝐵(𝑛, 𝑝) 𝑛 and 𝑝 are said to be parameter of binomial distribution.
Examples:
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𝑋~𝐵(8, 0.35) 𝑌~𝐵 (15, 3) 𝑀~𝐵(40, 0.1) 𝑁~𝐵(20, 0.7)
𝑍~𝐵(30, 0.85)
Formula of binomial distribution
𝑃(𝑋 = 𝑟) = 𝐶𝑟𝑛 𝑞𝑛−𝑟 𝑝𝑟
 𝑋 is idscrete random variable,
 𝑛 is total number of trials,
 𝑟 is the number of reuired outcomes out of n trials,
 𝑝 is probability of success,
 𝑞 is probability of failure?
Note: Binomial distribution is used when the size of population is not given, just probability are
given.
Q.1
On an occasion when a particular gymnast performs a certain routine, the probability
that she will perform it correctly is 0.65, independently of all other occasions. Find the
probability that she will perform the routine correctly on exactly 5 occasions out of 7.
Q.2
A manufacturer of biscuits produces 3 times cream biscuits as chocolate ones. Biscuits
are chosen randomly and packed into boxes of 10. Find the probability that a box contains equal
number of cream biscuits and chocolate biscuits.
Q.3
65% of all watches sold by a shop have a digital display and 35% has an analog display.
Find the probability that, out of next 12 customers who buy a watch, fewer than10 chooses one
with digital display.
Q.4
The random variable X is distributed 𝐵(7, 0.2). Find, correct to three decimal places,
i. 𝑃(𝑋 = 3),
ii. 𝑃(1 < 𝑋 ≤ 4),
iii. 𝑃(𝑋 > 3).
Expectation and variance of the Binomial Distribution: It can e shown that
If 𝑋~𝐵(𝑛, 𝑝) then the expected value of X is given by
𝐸(𝑋) = 𝑛𝑝.
and the variance of 𝑋 is given by
𝑉𝑎𝑟(𝑋) = 𝑛𝑝𝑞 Where 𝑞 = 𝑝 − 1
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Q.5
The probability that New Year “s Day is on Saturday in a randomly chosen year is 7.
15 years are chosen randomly. Find the probability that at least 3 of these years have
New Year’s Day on a Saturday.
Q.6
A survey of adults in a certain large town found that 76% of people wore a watch on
their left wrist, 15% wore a watch on their right wrist and 9 % did not wear watch. A
random sample of 14 people was taken. Find the probability that more than 2 adults did
not wear a watch.
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
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Q.7
The mean number of defective batteries in packs of 20 is 1.6. Use a binomial distribution
to calculate the probability that a randomly chosen pack of 20 will have more than 2
defective batteries.
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
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