In a sample space 𝑆, the probabilities are shown for the combinations of events 𝐴 and 𝐵 occurring. Are 𝐴 and 𝐵 independent events?
What can we say about the probability of independent events? The probability of 𝐴 and 𝐵 occurring equals the probability of 𝐴 times the probability of 𝐵. We need to check if the probability of 𝐴 and 𝐵, the probability in the middle, is equal to the probability of 𝐴 times the probability of 𝐵.
But before we do that, notice how each of these probabilities have a different denominator. In order to work with these and compare them accurately, we need to have a common denominator. The least common multiple from five, 10, and 15 will be 30. To go from 15 to 30, we’ll multiply by two. If we multiply by two in our denominator, we need to multiply by two in our numerator. Seven times two equals 14. Five times six equals 30. If we multiply by six in the denominator, we need to multiply by six in the numerator. One times six equals six. 10 times three equals 30. And one times three equals three.
Seventeen [seven] fifteenths equals fourteen thirtieths. One-fifth equals six thirtieths. And one-tenth equals three thirtieths. The probability of 𝐴 and 𝐵 will be the intersection of the probabilities 𝐴 and 𝐵. That’s six thirtieths.
The probability of 𝐴 is a little bit trickier. The probability of 𝐴 is equal to fourteen thirtieths plus six thirtieths. It’s equal to the two probabilities found in the 𝐴 circle. Fourteen thirtieths plus six thirtieths equals twenty thirtieths. And we need to multiply that by the probability of event 𝐵. Event 𝐵 will be six thirtieths plus three thirtieths, both of the probabilities found in the 𝐵 circle. Adding six thirtieths and three thirtieths, we get nine thirtieths.
To multiply fractions, we multiply their numerators. 20 times nine equals 180. 30 times 30 equals 900. We can simplify by dropping those zeros. And then, we could say that 90 is divisible by 18. We can divide the numerator and denominator by 18. 18 divided by 18 equals one. 90 divided by 18 equals five.
We now need to ask the question is six thirtieths equal to one-fifth? Remember that we already calculated what one-fifth is. We know that one-fifth equals six thirtieths.
And because the probability of 𝐴 and 𝐵 is equal to the probability of 𝐴 times the probability of 𝐵, we can say that, yes, these are independent events.