Probability Calculation: Odds Against Happening

The question asks us to find the "odds against happening" for an event A, given that the probability of A happening is $P(A) = \frac{1}{3}$.

Understanding Odds Against

The odds against an event are defined as the ratio of the probability that the event will *not* occur to the probability that the event *will* occur.

Mathematically, this is expressed as:

Odds Against Event = P(Event Not Happening) : P(Event Happening)

Or, using notation:

Odds Against A = P(\text{not } A) : P(A)

Calculating the Probability of Not Happening

First, we need to find the probability that event A does *not* happen. We know that the sum of the probability of an event happening and the probability of it not happening is always 1.

So, $P(\text{not } A) = 1 - P(A)$.

Given $P(A) = \frac{1}{3}$, we can calculate $P(\text{not } A)$:

$$P(\text{not } A) = 1 - \frac{1}{3}$$

To subtract, we find a common denominator:

$$P(\text{not } A) = \frac{3}{3} - \frac{1}{3}$$

$$P(\text{not } A) = \frac{2}{3}$$

Determining the Odds Against

Now we can use the formula for odds against:

Odds Against A = P(\text{not } A) : P(A)

Substitute the values we have:

Odds Against A = \frac{2}{3} : \frac{1}{3}

To simplify this ratio, we can multiply both parts of the ratio by the common denominator, which is 3:

Odds Against A = \left(\frac{2}{3} \times 3\right) : \left(\frac{1}{3} \times 3\right)

Odds Against A = 2 : 1

Conclusion

The odds against the happening of event A are 2 : 1. This means that for every 2 times the event is expected *not* to happen, it is expected to happen 1 time.

Comparing this result with the given options, the correct option is 2 : 1.