34. Two students (say A, and B) are challenged to solve a problem in statistics. Suppose that
the probability that A will solve the problem is 40%, the probability that B will solve the
problem is 55%, and the probability that A or B will solve the problem is 25%. The
probability that the problem will be solved (that is both A and B will solve the problem) is
(a) 70%
(b) 95%
(c) 30%
(d) 75%
35. Smita has 88% chance of receiving an A grade in statistics; a 45% chance of receiving an A
in physics, and a 87% chance of receiving an A in either statistics or physics. The
probability that she will receive an A in both statistics and physics:
(a) 70%
(b) 46%
(c) 13%
(d) 50%
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Let's solve the problems step by step:
For problem 34:
Given:
Probability that A solves the problem = 40% = 0.4
Probability that B solves the problem = 55% = 0.55
Probability that A or B solves the problem = 25% = 0.25
To find:
Probability that both A and B solve the problem.
We can use the formula:
P(A or B) = P(A) + P(B) - P(A and B)
Let's substitute the given values into the formula:
0.25 = 0.4 + 0.55 - P(A and B)
Let's rearrange the equation to solve for P(A and B):
P(A and B) = 0.4 + 0.55 - 0.25
P(A and B) = 0.7
Therefore, the probability that both A and B solve the problem is 70%.
So, the answer to problem 34 is (a) 70%.
For problem 35:
Given:
Probability of receiving an A in statistics = 88% = 0.88
Probability of receiving an A in physics = 45% = 0.45
Probability of receiving an A in either statistics or physics = 87% = 0.87
To find:
Probability of receiving an A in both statistics and physics.
We can use the formula:
P(A and B) = P(A) + P(B) - P(A or B)
Let's substitute the given values into the formula:
P(A and B) = 0.88 + 0.45 - 0.87
P(A and B) = 0.46
Therefore, the probability of receiving an A in both statistics and physics is 46%.
So, the answer to problem 35 is (b) 46%.
To solve these problems, we can use the concept of probabilities and set up equations based on the given information. Let's break down each problem:
34. The problem asks for the probability that both students, A and B, will solve the problem. We are given the individual probabilities for A and B, as well as the probability that either A or B will solve it.
Let's assume that event A represents the probability of A solving the problem, event B represents the probability of B solving the problem, and event C represents the probability of either A or B solving the problem.
We are given:
P(A) = 40% = 0.4
P(B) = 55% = 0.55
P(A or B) = 25% = 0.25
To find the probability that both A and B will solve the problem, we can use the formula:
P(A and B) = P(A) + P(B) - P(A or B)
Substituting the given values:
P(A and B) = 0.4 + 0.55 - 0.25
P(A and B) = 0.7
Therefore, the probability that both A and B will solve the problem is 70%. Hence, the answer is (a) 70%.
35. The problem asks for the probability that Smita will receive an A grade in both statistics and physics. We are given the individual probabilities for statistics, physics, and the probability of receiving an A in either subject.
Let's assume that event S represents the probability of receiving an A in statistics, event P represents the probability of receiving an A in physics, and event E represents the probability of receiving an A in either subject (statistics or physics).
We are given:
P(S) = 88% = 0.88
P(P) = 45% = 0.45
P(S or P) = 87% = 0.87
To find the probability that Smita will receive an A in both statistics and physics, we can use the formula:
P(S and P) = P(S) + P(P) - P(S or P)
Substituting the given values:
P(S and P) = 0.88 + 0.45 - 0.87
P(S and P) = 0.46
Therefore, the probability that Smita will receive an A in both statistics and physics is 46%. Hence, the answer is (b) 46%.