Question

Let \(A\) and \(B\) be independent events such that \(P(A) = p, P(B) = 2p\). The largest value of \(p\), for which \(P(\text{exactly one of } A, B \text{ occurs}) = \frac{5{9}\), is :}

• A. \(\frac{1}{3}\)
• B. \(\frac{4}{9}\)
• C. \(\frac{5}{12}\)
• D. \(\frac{2}{9}\)

Hint:

The expression for "exactly one happens" is symmetric. Always verify that your final \(p\) values allow for \(2p \leq 1\), as probabilities cannot exceed 1. Here \(2(5/12) = 5/6<1\), so it is valid.

Answer 1

The Correct Option is C

To solve this problem, we need to determine the largest value of \( p \) for which the probability that exactly one of the events \( A \) or \( B \) occurs is equal to \( \frac{5}{9} \).

Given that \( A \) and \( B \) are independent events:

• \( P(A) = p \)
• \( P(B) = 2p \)
• The probability that exactly one of \( A \) or \( B \) occurs is \( P(A \cap \overline{B}) + P(\overline{A} \cap B) \).

Now let's calculate each part:

• \( P(A \cap \overline{B}) = P(A) \times P(\overline{B}) = P(A) \times (1 - P(B)) = p \times (1 - 2p) \).
• \( P(\overline{A} \cap B) = P(\overline{A}) \times P(B) = (1 - P(A)) \times P(B) = (1 - p) \times 2p \).

The expression for the probability that exactly one of \( A \) or \( B \) occurs is:

P(\text{exactly one of } A, B \text{ occurs}) = p(1 - 2p) + (1 - p)2p

Simplify this expression:

= p - 2p^2 + 2p - 2p^2
= 3p - 4p^2

We need this to equal \(\frac{5}{9}\):

3p - 4p^2 = \frac{5}{9}

Rearrange the equation:

4p^2 - 3p + \frac{5}{9} = 0

Multiply the entire equation by 9 to clear the fraction:

36p^2 - 27p + 5 = 0

This is a quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a = 36\), \(b = -27\), \(c = 5\).

To find \( p \), use the quadratic formula:

p = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}

Substitute the values of \(a\), \(b\), and \(c\):

p = \frac{{27 \pm \sqrt{{(-27)^2 - 4 \cdot 36 \cdot 5}}}}{2 \cdot 36}

Calculate the discriminant:

= \frac{{27 \pm \sqrt{{729 - 720}}}}{72} = \frac{{27 \pm \sqrt{9}}}{72} = \frac{{27 \pm 3}}{72}

This gives two possible solutions:

p = \frac{30}{72} = \frac{5}{12} and p = \frac{24}{72} = \frac{1}{3}.

Among these, the largest value of \( p \) is \frac{5}{12}.

Thus, the largest value of \( p \) for which the given condition holds true is \frac{5}{12}.