Probability and independence

Question

I am trying to solve a particular probability question.

I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number.

I find the probability as:

P(multiple of 5) OR P(odd number)=P(multiple of 5) + P(odd number)-[P(multiple of 5) AND P(odd number)]=2/10+5/10-[(2/10)(5/10)]=6/10 (which is the correct answer)

Notice that I used the assumption that rolling a multiple of 5 is independent of rolling an odd number, since I essentially multiplied their probabilities to get the answer. However, rolling a multiple of 5 INVOLVES rolling an odd number in one case, namely rolling a 5.

So did I get the correct answer by fluke? Can the multiplication rule also be used to describe non-independent events sometimes?

Answer 1

No, it's not a fluke, in the sense that $P(A\cap B)=P(A)P(B)$ is the definition of independence of $A$ and $B$, so if it holds, as it does here, $A$ and $B$ are necessarily independent. In another sense, it was a fluke, in that it might not have been obvious that the events multiple of 5 and odd are independent: this depends on the fact that there are exactly as many odd as non-odd multiples of 5 between 1 and 10.

So you got lucky, this time: your two events really are independent. What would happen if you'd been asked about a 9-sided die? Or a 10-sided die, but with the events multiple of 3 and even?

In general, arithmetic progressions like $(na+c)_{n\in\Bbb{N}}$ and $(nb+d)_{n\in\Bbb{n}},$ e.g. odds when $a=c=1$ or multiples of 5 when $b=5,c=0,$ are "independent" in $\Bbb{N}$ in the sense of natural density whenever $a$ and $b$ are relatively prime. To bring this fact down to the finite realm of dice with $k$ sides, we need $k$ to be a common multiple of $a$ and $b$. In fact 10 is a common multiple of 2 and 5, and that indicates a bigger reason why you lucked out with this computation.

Answer 2

Its P(multiple of 5) + P(odd) - P(multiple of 5 and odd). It's not mutually exclusive. The two events can happen at the same time, just like a the fact that it can rain when it is sunny. 2/10 + 5/10 - 1/10 = 3/5

Answer 3

It might seem odd, but in fact the two events of rolling a multiple $5$ and rolling an odd number are independent.

To see this, let $A$ be the event that a multiple of $5$ is rolled, and $B$ be the event that an odd number is rolled. What is $P(A)$? What is $P(A|B)$? Since the two numbers are the same, it follows that the outcome of $B$ does not affect that of $A$, so the events are independent.

As long as events $A$ and $B$ are independent, then $P(A\text{ or }B)=P(A)+P(B)-P(A)P(B)$, so you figured correctly. The only part you were missing is proving that $A$ and $B$ are in fact independent.

Of course, the easiest way to solve this problem is to just list possibilities. $6$ of the $10$ sides are odd numbers or multiples of $5$, so the probability is $6/10$.

Finally, I should mention that the multiplication rule only holds for independent events, because one way to define independence is by the formula $P(A\text{ and }B)=P(A)P(B).$

Answer 4

Let $A$ be the event we roll an odd number, and $B$ the event we roll a multiple of $5$, The events $A$ and $B$ are independent. For it is clear that the probability of $A\cap B$ is $\frac{1}{10}$, and the product $\Pr(A)\Pr(B)$ is also $\frac{1}{10}$.

However, consider a $9$-sided die. Then the probability of getting a multiple of $5$ and an odd number is $\frac{1}{9}$. The probability of rolling a $5$ is $\frac{1}{9}$, and the probability of odd is $\frac{5}{9}$. The product is $\frac{5}{81}$, which is quite different from $\frac{1}{9}$.

Thus the $10$-sided die is peculiar for this problem. In general, there is no reason to expect independence. A calculation based on multiplying is in principle incomplete unless one proves independence. And for most problems of this general type, we do not have independence.

Answer 5

Actually when we through a fair die each outcome is identically independently distributed, so there is no need to assume multiple of 5 is independent of rolling an odd number. Two events are said to be dependent if they effect occurrence or non-occurrence of other event. In your case both events are Independent.