We say that two events are independent if the occurrence of one event does not affect the probability of the other occurring.
For example,
To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and then multiply the probabilities. This multiplication rule is defined symbolically as
$\text{P(A and B)}$P(A and B)$=$=$P(A$P(A$B)$B)$=$=$P(A)$P(A) x $P(B)$P(B), for independent events. The symbol that looks like a shoehorn is used in the case where we have one event AND another occurring.
This means you can also test for independence by verifying if P(A) x P(B) = P(A and B).
A coin is tossed and a single 6-sided die is rolled. Find the probability of flipping a tail on the coin and rolling a 4 on the die.
$\text{P(Tail) }=\frac{1}{2}$P(Tail) =12
$\text{P(4) }=\frac{1}{6}$P(4) =16
$\text{P(Tail and 4) }$P(Tail and 4) | $=$= | $\text{P(Tail) }\times\text{P(4) }$P(Tail) ×P(4) |
$=$= | $\frac{1}{2}\times\frac{1}{6}$12×16 | |
$=$= | $\frac{1}{12}$112 |
A nationwide survey found that $64%$64% of people in a small country town have unreliable internet access. If 3 people are selected at random, what is the probability that all three have unreliable internet access?
P(I) x P(I) x P(I) = $0.64\times0.64\times0.64=0.262144$0.64×0.64×0.64=0.262144
Now why are the events independent here? You may think that in the case of selecting people from a population, we should not replace the first person before selecting the next. This would make the selections dependent. But consider the following:
Say the population is $1000000$1000000, and $640000$640000 of them do not have reliable internet access.
P(first person has unreliable internet access) = $\frac{640000}{1000000}=0.64$6400001000000=0.64
If we remove one of these people from the population before the second draw, there would be $999999$999999 people left in the population, and $639999$639999 of them would have unreliable internet access:
P(second person has unreliable internet access) = $\frac{639999}{999999}=0.639$639999999999=0.639..
Without going any further, you can see that the probability does not change that much.
So for a large sample space, the probability changes so little that we can consider successive events as being independent.
A coin is tossed twice.
Construct a tree diagram showing the results of the given experiment.
Use the tree diagram to find the probability of getting exactly 1 Head.
Use the tree diagram to find the probability of getting 2 heads.
Use the tree diagram to find the probability of getting no heads.
Use the tree diagram to find the probability of getting 1 Head and 1 Tail.
Christa enters a competition in which she guesses the 3-digit code (from 000 to 999) which cracks open a vault containing one million dollars. If the 3-digit number to open the vault is randomly generated by a computer, what is the probability that it is:
an odd number?
an even number (including 000)?
a number greater than 123?
a number divisible by 10?
a number less than 321?
Two dice are rolled, and the numbers appearing on the uppermost faces are added. Find:
P(a total of at least 7)