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Chapter 1: Problem 8

If \(P(E)=0.9\) and \(P(F)=0.8\), show that \(P(E F) \geqslant 0.7\). In general, show that $$ P(E F) \geqslant P(E)+P(F)-1 $$ This is known as Bonferroni's inequality.

Short Answer

Expert verified
In general, Bonferroni's inequality states that \( P(E 鈭� F) \geq P(E) + P(F) - 1 \). Given the probabilities \( P(E) = 0.9 \) and \( P(F) = 0.8 \), by applying this inequality, we find that \( P(E 鈭� F) \geq 0.9 + 0.8 - 1 = 0.7 \). Hence, we have shown that \( P(E 鈭� F) \geq 0.7 \), verifying the truth of Bonferroni's inequality in this case.

Step by step solution

01

Understand the problem

We need to show that the probability of the intersection of two events is greater than or equal to the sum of their probabilities minus 1. This is known as Bonferroni's inequality.
02

Apply the formula for the probability of the union of two events

The probability of the union of two events E and F is given by: \( P(E 鈭� F) = P(E) + P(F) - P(E 鈭� F) \)
03

Rearrange the formula to solve for the intersection probability

We can rearrange the formula from Step 2 to solve for the probability of the intersection of events E and F: \( P(E 鈭� F) = P(E) + P(F) - P(E 鈭� F) \)
04

Apply the boundaries of the probabilities to conclude the inequality

Given that the probability of any event is between 0 and 1, we can conclude that the minimum value for P(E 鈭� F) equals 0, and the maximum value equals 1. Thus with the minimum value of 0, \( P(E 鈭� F) = P(E) + P(F) - 0 \) And with the maximum value of 1, \( P(E 鈭� F) \geq P(E) + P(F) - 1 \)
05

Verify with the given probabilities P(E) and P(F)

Now we can check the given probabilities to see if the inequality holds: \( P(E) = 0.9, P(F) = 0.8 \) According to the inequality from step 4: \( P(E 鈭� F) \geq 0.9 + 0.8 - 1 = 0.7 \) Thus, we have shown that the option \( P(E鈭〧) \geq 0.7 \) holds, and in general Bonferroni's inequality is true.

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Most popular questions from this chapter

In an election, candidate \(A\) receives \(n\) votes and candidate \(B\) receives \(m\) votes, where \(n>m .\) Assume that in the count of the votes all possible orderings of the \(n+m\) votes are equally likely. Let \(P_{n, m}\) denote the probability that from the first vote on \(A\) is always in the lead. Find (a) \(P_{2,1}\) (b) \(P_{3,1}\) (c) \(P_{n, 1}\) (d) \(P_{3,2}\) (e) \(P_{4,2}\) (f) \(P_{n, 2}\) (g) \(P_{4,3}\) (h) \(P_{5,3}\) (i) \(P_{5,4}\) (j) Make a conjecture as to the value of \(P_{n, m}\).

If the occurrence of \(B\) makes \(A\) more likely, does the occurrence of \(A\) make \(B\) more likely?

If two fair dice are tossed, what is the probability that the sum is \(i, i=2,3, \ldots, 12 ?\)

Let \(E, F, G\) be three events. Find expressions for the events that of \(E, F, G\) (a) only \(F\) occurs, (b) both \(E\) and \(F\) but not \(G\) occur, (c) at least one event occurs, (d) at least two events occur, (e) all three events occur, (f) none occurs, (g) at most one occurs, (h) at most two occur.

Suppose that 5 percent of men and \(0.25\) percent of women are color-blind. A colorblind person is chosen at random. What is the probability of this person being male? Assume that there is an equal number of males and females.
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