class 12

Math

Algebra

Probability I

Four person independently solve a certain problem correctly with probabilities $21 ,43 ,41 ,81 ˙$Then the probability that he problem is solve correctly by at least one of them is$256235 $b. $25621 $c. $2563 $d. $256253 $

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Five persons entered the lift cabin on the ground floor of an eight-floor house. Suppose that each of them, independently and with equal probability, can leave the cabin at any floor beginning with the first. Find out the probability of all five persons leaving at different floors.

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, then find the probability that it is rusted or is a nail.

Eight players $P_{1},P_{2},…,P_{8}$ play a knock-out tournament. It is known that whenever the players $P_{i}$ and $P_{j}$ play, the player $P_{i}$ will win if $i<j$. Assuming that the players are paired at random in each round, what is the probability that the player $P_{4}$ reaches the final?

A cube having all of its sides painted is cut by two horizontal, two vertical, and other two planes so as to form 27 cubes all having the same dimensions. Of these cubes, a cube is selected at random. The probability that the cube selected has none of its sides painted is

In a class of 125 students 70 passed in Mathematics, 55 in Statistics, and 30 in both. Then find the probability that a student selected at random from the class has passed in only one subject.

Let A be a set containing n elements A subset P of the set A is chosen at random. The set A is reconstructed by replacing the elements of P, and another subset Q of A is chosen at random. The probability that $P∩Q$ contains exactly $m(m<n)$ elements is

A dice is loaded so that the probability of a face I is proportional to I, I = 1, 2, …, 6. Then find the probability of an even number occurring when the dice is rolled.

There are two die A and B both having six faces. Die A has three faces marked with 1, two faces marked with 2, and one face marked with 1, two faces marked with 2, and one face marked with 3. Die B has one face marked with 1, two faces marked with 2, and three faces marked with 3. Both dices are thrown randomly once. If E be the event of getting sum of the numbers appearing on top faces equal to x and let P(E) be the probability of event E, then When x = 4, then P(E) is equal to