Independent events

What you will be able to do

• test whether two events are independent from their probabilities or from a table;
• explain why independent events are not the same as mutually exclusive events;
• use the product rule for several independent events, including series and parallel systems.

Two tosses of a coin

Toss a coin and get a head. Toss it again. Does the first head change the chance of a head on the second toss? It does not. The coin has no memory. The two tosses are independent.

Now take two cards from a pack without replacement. If the first card is an ace, fewer aces remain, and the chance of an ace on the second card goes down. These two draws are dependent.

The definition

Events $A$ and $B$ are independent when

This is the same as saying $P(A \mid B) = P(A)$ when $P(B) > 0$: the news that $B$ happened leaves the probability of $A$ where it was. Put the product rule into the multiplication rule and you see the two statements agree.

To test independence, work out three numbers: $P(A)$, $P(B)$ and $P(A \cap B)$. Then compare $P(A \cap B)$ with the product.

A test with two dice

Two fair dice are thrown. Let $A$ be "the first die shows $3$" and $B$ be "the sum is $7$". Then $P(A) = \dfrac{1}{6}$ and $P(B) = \dfrac{6}{36} = \dfrac{1}{6}$. Only $(3, 4)$ is in both, so $P(A \cap B) = \dfrac{1}{36}$. The product $\dfrac{1}{6} \times \dfrac{1}{6}$ is $\dfrac{1}{36}$. So $A$ and $B$ are independent, which many students find surprising.

Change $B$ to "the sum is $8$". Now $P(B) = \dfrac{5}{36}$ and $P(A \cap B) = \dfrac{1}{36}$, but the product is $\dfrac{1}{6} \times \dfrac{5}{36} = \dfrac{5}{216}$. These differ, so the events are dependent.

Independent is not the same as mutually exclusive

Events are mutually exclusive when they cannot happen together, so $A \cap B = \varnothing$. That is a statement about overlap. Independence is a statement about information.

If $P(A) > 0$ and $P(B) > 0$ and the events are mutually exclusive, then $P(A \cap B) = 0$, but the product $P(A)\,P(B)$ is not zero. So mutually exclusive events with positive probabilities are never independent. Knowing that $A$ happened tells you that $B$ did not.

Complements and several events

If $A$ and $B$ are independent, so are $A$ and $B'$, and so are $A'$ and $B'$. For the first pair:

For three events, mutual independence needs four conditions: the product rule for each of the three pairs, and also $P(A \cap B \cap C) = P(A)\,P(B)\,P(C)$. Two fair coins show why. Let $A$ be "the first is a head", $B$ be "the second is a head" and $C$ be "both coins agree". Each pair is independent, because each has probability $\dfrac{1}{2}$ and each pair meets with probability $\dfrac{1}{4}$. But $A \cap B \cap C$ is just HH, with probability $\dfrac{1}{4}$, while the product of the three is $\dfrac{1}{8}$. The events are pairwise independent but not mutually independent.

A worked example with two sensors

A machine has two sensors that work independently. One works with probability $0.9$ and the other with probability $0.8$.

1. Both work. $0.9 \times 0.8 = 0.72$. If the machine needs both, this is the series case.
2. At least one works. The chance that both fail is $0.1 \times 0.2 = 0.02$. So at least one works with probability $1 - 0.02 = 0.98$. This is the parallel case.
3. Exactly one works. $0.9 \times 0.2 + 0.1 \times 0.8 = 0.18 + 0.08 = 0.26$. Check: $0.72 + 0.26 + 0.02 = 1$.

Practise with new numbers

Try it in Python

The program tests the product rule for pairs of events on two dice, using exact fractions so that there is no rounding.

Common mistakes

• Assuming independence. Draws without replacement are dependent. Use independence only when the situation says so, or when the test passes.
• Mixing up independent and mutually exclusive. Mutually exclusive events with positive probabilities are always dependent.
• Adding for "and". For independent events, "and" means multiply. Adding gives a probability that may exceed $1$.
• Checking only pairs. Three events need the triple product as well.

Remember these

• $A$ and $B$ are independent when $P(A \cap B) = P(A)\,P(B)$, or equivalently $P(A \mid B) = P(A)$.
• Independent events: the complements are independent too.
• Mutually exclusive events with positive probabilities are never independent.
• Series: $p^n$. Parallel: $1 - (1 - p)^n$, for $n$ independent components that each work with probability $p$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Events $A$ and $B$ are independent, with $P(A) = 0.4$ and $P(B) = 0.5$. Find $P(A \cap B)$ and $P(A \cup B)$.

2. For two events, $P(A) = 0.5$, $P(B) = 0.3$ and $P(A \cap B) = 0.15$. Are they independent?

3. Events $A$ and $B$ are mutually exclusive, with $P(A) = 0.3$ and $P(B) = 0.4$. Can they be independent?

4. A fair coin is tossed $3$ times. Find the probability of at least one tail.

5. Events $A$ and $B$ are independent with $P(A) = 0.7$ and $P(B) = 0.2$. Find $P(A \cap B')$.

References

• OpenStax, Introductory Statistics, Chapter 3, section on independent and mutually exclusive events.
• NCERT, Mathematics Part II for Class XII, Chapter 13, section on independent events.