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Binomial Distribution Conditions of binomial distribution: There are four basic conditions for binomial distribution. i. A finite number, π , 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, π, of a successful outcome is the same for each trial. The discrete random variable, π, is the number of successful outcomes in n trials. If above four conditions are satisfied by the set π then π is said to follow the binomial distribution. It is written as π~π΅(π, π) π and π are said to be parameter of binomial distribution. Examples: 1 π~π΅(8, 0.35) π~π΅ (15, 3) π~π΅(40, 0.1) π~π΅(20, 0.7) π~π΅(30, 0.85) Formula of binomial distribution π(π = π) = πΆππ ππβπ ππ ο· π is idscrete random variable, ο· π is total number of trials, ο· π is the number of reuired outcomes out of n trials, ο· π is probability of success, ο· π 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 π΅(7, 0.2). Find, correct to three decimal places, i. π(π = 3), ii. π(1 < π β€ 4), iii. π(π > 3). Expectation and variance of the Binomial Distribution: It can e shown that If π~π΅(π, π) then the expected value of X is given by πΈ(π) = ππ. and the variance of π is given by πππ(π) = πππ Where π = π β 1 1 Q.5 The probability that New Year βs 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 Yearβs 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 1 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 2