Understanding the Problem

We are given two independent events E and F. We know:

• Probability that exactly one of them occurs is $\frac{11}{25}$.
• Probability that none of them occurs is $\frac{2}{25}$.

We need to find the individual probabilities P(E) and P(F).

Step-by-Step Solution

Step 1: Define the Probabilities

Let $p=P\left(E\right)$ and $q=P\left(F\right)$.

Since the events are independent, the probability of both occurring is $P(E\cap F)=P\left(E\right)·P\left(F\right)=pq$.

Step 2: Probability of None Occurring

The probability that an event does NOT occur is 1 minus the probability that it occurs.

Therefore, the probability that neither E nor F occurs is:

$P(\overline{E}\cap \overline{F})=P\left(\overline{E}\right)·P\left(\overline{F}\right)=(1-p)(1-q)$

We are told this equals $\frac{2}{25}$.

This gives us our first equation:

Equation 1: $(1-p)(1-q)=\frac{2}{25}$

Step 3: Probability of Exactly One Occurring

The event "exactly one occurs" can happen in two mutually exclusive ways:

1. E occurs and F does not occur: $P(E\cap \overline{F})=p(1-q)$
2. F occurs and E does not occur: $P(\overline{E}\cap F)=(1-p)q$

The total probability is the sum of these two:

$p(1-q)+(1-p)q=\frac{11}{25}$

We can simplify the left side:

$p-pq+q-pq=p+q-2pq$

This gives us our second equation:

Equation 2: $p+q-2pq=\frac{11}{25}$

Step 4: Solve the System of Equations

Let's expand Equation 1:

$(1-p)(1-q)=1-p-q+pq=\frac{2}{25}$

This can be rewritten as:

$p+q-pq=1-\frac{2}{25}=\frac{23}{25}$

Let's call this Equation 1a:

$p+q-pq=\frac{23}{25}$

Now we have two equations:

Equation 1a: $p+q-pq=\frac{23}{25}$

Equation 2: $p+q-2pq=\frac{11}{25}$

Subtract Equation 2 from Equation 1a:

$(p+q-pq)-(p+q-2pq)=\frac{23}{25}-\frac{11}{25}$

$pq=\frac{12}{25}$

Now we know the product of p and q. Substitute $pq=\frac{12}{25}$ back into Equation 1a:

$p+q-\frac{12}{25}=\frac{23}{25}$

$p+q=\frac{23}{25}+\frac{12}{25}=\frac{35}{25}=\frac{7}{5}$

So now we have:

Sum: $p+q=\frac{7}{5}$

Product: $pq=\frac{12}{25}$

This means p and q are the roots of the quadratic equation:

$x^2-(p+q)x+pq$