Two-step experiments are those that incorporate two simple experiments, for example tossing a coin and rolling a die, or tossing a coin twice. Finding probabilities of two-step experiments is easier if we use a list, table, or tree diagram to show all possible outcomes.
A table is useful for showing all possible outcomes of two events in the rows and columns.
1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|
\text{H} | \text{H}1 | \text{H}2 | \text{H}3 | \text{H}4 | \text{H}5 | \text{H}6 |
\text{T} | \text{T}1 | \text{T}2 | \text{T}3 | \text{T}4 | \text{T}5 | \text{T}6 |
Each cell in the table is an outcome of rolling a die and a coin. There are 12 possible outcomes in the sample space.
A player is rolling 2 dice and looking at their sum. They draw up a table of all the possible dice rolls for two dice and what they sum to.
1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 |
2 | 3 | 4 | 5 | 6 | 7 | 8 |
3 | 4 | 5 | 6 | 7 | 8 | 9 |
4 | 5 | 6 | 7 | 8 | 9 | 10 |
5 | 6 | 7 | 8 | 9 | 10 | 11 |
6 | 7 | 8 | 9 | 10 | 11 | 12 |
What is the probability the dice will sum to 8?
A table is useful for showing all possible outcomes of two events in the rows and columns.
A tree diagram is useful in tracking two-step experiments. It is named because the diagram that results looks like a tree.
It is useful if the events have different weightings or are unequal events.
When the outcomes are not equally likely the probability will be written on the branches.
The sum of the probabilities on branches from a single node should sum to 1.
When a single trial is carried out, we have just one column of branches.
Here are some examples. None of these have probabilities written on the branches because the outcomes are equally likely.
Here are some examples that have probabilities on the branches, because they do not have an equal chance of occurring.
When looking at a group of branches that come from a single point, the sum of the group always adds to 1 (or 100\%). This indicates that all the outcomes are listed.
When more than one experiment is carried out, we have two (or more) columns of branches.
To find the probability of at least 1 win, we could do either
a) \text{P(win, lose)} + \text{P(lose, win)} + \text{P(win, win)} = 21\% + 21\% + 9\% = 51\% or
b) Use the complementary event of losing both games and calculate:
1 - \text{P (lose, lose)} = 1- 49\% = 51\%
For multistage events where the next stage is affected by the previous stage, we call these dependent events. We need to take care when drawing the tree diagram accordingly.
One type of experiment that is dependent on previous trials is an experiment without replacement. This means that the object selected (e.g. card, marble, person) is not able to be selected in the second selection.
Han owns four green ties and three blue ties. He selects one of the ties at random for himself and then another tie at random for his friend.
Write the probabilities for the outcomes on the edges of the probability tree diagram.
What is the probability that Han selects a blue tie for himself?
Calculate the probability that Han selects two green ties
A tree diagram is useful in tracking two-step experiments. It is named because the diagram that results looks like a tree.
For two-step experiments:
Multiply along the branches to calculate the probability of individual outcomes.
Add down the list of outcomes to calculate the probability of multiple options.
The final percentage should add to 100, or the final fractions should add to 1 - this is useful to see if you have calculated everything correctly.