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Here are a few recommended readings before getting started with this lesson.
Two events $A$ and $B$ are independent events if the occurrence of either of these events does not affect the occurrence of the other. It is also said that they are independent if and only if the probability that both events occur is equal to the product of the individual probabilities.
Two events $A$ and $B$ are considered dependent events if the occurrence of either of these events affects the occurrence of the other. If the events are dependent, the probability that both events occur is equal to the product of the probability of the first event occurring and the probability of the second event occurring after the first event.
As mentioned above, this principle holds true only if the events are independent of each other. If the events are dependent, multiplying the number of possible outcomes for each event will not be correct. Considering the previous example, suppose now that the spiral-bound notebooks came only in red.
There are still $2$ types of notebooks and a total of $3$ colors for the non spiral bound notebooks. However, the possible number of different notebooks a customer may buy is not $2×3=6.$ Rather, it is $4.$ This happens because, in this case, the possible colors for a notebook depend on the type of notebook.
While preparing the raffle, Tiffaniqua considered inviting everyone who buys a ticket to roll a die and toss a coin.
Fundamental Counting Principle |
If an event $A$ has $n$ possible outcomes and an event $B$ has $m$ possible outcomes, then the total number of different outcomes for $A$ and $B$ combined is $n⋅m.$ |
According to the tree diagram, there are $12$ possible outcomes. This is the same number as the one found through the Fundamental Counting Principle. Tiffaniqua wanted to have more possible prizes, so she had to come up with an idea for the raffle with more possible outcomes!
A permutation is an arrangement of objects in which the order is important. For example, consider constructing a number using only the digits $4,$ $5,$ and $6$ without repetitions. Any of the three digits can be picked for the first position, leaving two choices for the second position, then only one choice for the third position.
In this case, there are six possible permutations.Tiffaniqua chose $5$ prizes to display at her stall — a cinema ticket, a CD, a basketball, a teddy bear, and a pair of sunglasses. She is sure they will attract many people to her raffle!
Now all she has to do is decide in which order she should line the prizes up.
First Prize | $5$ choices |
---|---|
Second Prize | |
Third Prize | |
Fourth Prize | |
Fifth Prize |
When choosing the second prize, there are only $4$ possible outcomes. This is because one of the five prizes has already been chosen for the first prize.
First Prize | $5$ choices |
---|---|
Second Prize | $4$ choices |
Third Prize | |
Fourth Prize | |
Fifth Prize |
For the same reason, there are $3$ choices for the third prize, $2$ choices for the fourth prize, and only $1$ choice for the fifth prize.
First Prize | $5$ choices |
---|---|
Second Prize | $4$ choices |
Third Prize | $3$ choices |
Fourth Prize | $2$ choices |
Fifth Prize | $1$ choice |
First Prize | Teddy Bear |
---|---|
Second Prize | |
Third Prize | CD |
Fourth Prize | |
Fifth Prize |
The second prize could be the sunglasses, the basketball, or the cinema ticket. There are $3$ options for the second prize.
First Prize | Teddy Bear |
---|---|
Second Prize | $3$ options |
Third Prize | CD |
Fourth Prize | |
Fifth Prize |
The fourth prize cannot be the teddy bear, the CD, or the second prize. This means that there are $2$ options left for the fourth prize. Similarly, there is only $1$ option left for the fifth prize after the fourth prize is chosen.
First Prize | Teddy Bear |
---|---|
Second Prize | $3$ options |
Third Prize | CD |
Fourth Prize | $2$ options |
Fifth Prize | $1$ option |
Right before the event, a new sponsor decided to give Tiffaniqua a bunch of new prizes for her raffle! It was too late to change the way the raffle works to accommodate the new prizes. Instead, Tiffaniqua decided to have a side raffle!
When someone wants to take part in the side raffle, they draw a random prize from a box of $20$ prizes. The big box contains prizes like a pair of gloves and a pair of socks.