Probability Tes...

It is known that \(10 \%\) of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective?

A factory has two machines \(A\) and \(B\). Past record shows that machine A produced \(60 \%\) of the items of output and machine B produced \(40 \%\) of the items. Further, \(2 \%\) of the items produced by machine A and \(1 \%\) produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?

A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

If \(A\) and \(B\) are two events such that \(P ( A \cup B )\)= \(\frac{5}{6}\) , \(P ( A \cap B )\) = \(\frac{1} {3}\), \(P ( B )\) = \(\frac{1}2\), then the events \(A\) and \(B\) are:

Find the probability of getting 5 exactly twice in 7 throws of a die.

There are \(5 \%\) defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?

A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

A fair coin is tossed independently four times. The probability of the event "the number of times heads show up is more than the number of times tails show up" is:

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins/loses.

Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail', given that at least one die shows as \(3\).

How many times must a man toss a fair coin so that the probability of having at least one head is more than \(90 \%\)?

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black:

Ten eggs are drawn successively, with replacement, from a lot containing \(10 \%\) defective eggs. Find the probability that there is at least one defective egg.

A pair of dice is thrown. Find the probability of obtaining a sum of 8 or getting an even number on both the dice.

A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

A die is tossed thrice. Find the probability of getting an odd number at least once:

If a leap year is selected at random, what is the chance that it will contain 53 Tuesdays?

An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.

On a multiple-choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

A manufacturer has three machine operators \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\). The first operator A produces \(1 \%\) defective items, whereas the other two operators \(B\) and \(C\) produce \(5 \%\) and \(7 \%\) defective items respectively. \(A\) is on the job for \(50 \%\) of the time, \(B\) is on the job for \(30 \%\) of the time and \(C\) is on the job for \(20 \%\) of the time. A defective item is produced, what is the probability that it was produced by A?

\(A , B\) and \(C\) are three mutually exclusive and exhaustive events. \(P ( A )=2 P ( B )=6 P ( C )\), Find \(P ( B )\):

Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are \(0.6\) and \(0.4\) respectively. Further, if the first group wins, the probability of introducing a new product is \(0.7\) and the corresponding probability is \(0.3\) if the second group wins. Find the probability that the new product introduced was by the second group.

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale:

An urn contains 5 red and 5 black balls. A ball is drawn at random, its color is noted, and is returned to the urn. Moreover, 2 additional balls of the color drawn are put in the urn and then a ball is drawn at random. What is the probability that the second ball is red?

Suppose a girl throws a die. If she gets a 5 or 6 , she tosses a coin three times and notes the number of heads. If she gets \(1,2,3\) or 4 , she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw \(1,2,3\) or 4 with the die?

\(60 \%\) of the employees of a company are college graduates. Of these, \(10 \%\) are in sales. Of the employees who did not graduate from college, \(80 \%\) are in sales. The probability that an employee selected at random is in sales is:

Assume that the chances of a patient having a heart attack are \(40 \%\). It is also assumed that a meditation and yoga course reduce the risk of heart attack by \(30 \%\) and prescription of certain drug reduces its chances by \(25 \%\). At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.