Probability: Trees, Venn Diagrams and Conditional Events

Working out chance

Probability is the mathematics of how likely things are. It runs from the impossible to the certain, and it lets you reason clearly about uncertainty. This module works through it: - The probability scale: from zero to one, impossible to certain.\n- Combining events: mutually exclusive and independent events.\n- Tree diagrams: working through combined events step by step.\n- Venn diagrams and conditional probability: sorting and updating chances. Statistics, such as averages and data diagrams, has its own module. Here the focus is entirely on probability.

Words for probability

Five terms you need before you use them. Learn what each one means.

Match each probability idea to what it describes

Mutually exclusive against independent

Two of the most important ideas in probability are easily muddled. Getting them straight tells you whether to add or to multiply.

How do you combine independent events?

For two independent events, how do you find the chance that both of them happen?

• Multiply the chance of each event together
• Add the two chances together
• Subtract one chance from the other
• Ignore one of the events

Tap the two independent situations

Tap the TWO situations where the two events are independent of each other.

• Tossing a coin twice
• Rolling a die, then rolling it again
• Drawing two cards without replacing the first
• Taking two sweets from a bag without replacing

Reading a tree diagram

A tree diagram lays out combined events so nothing is missed, and it turns a wordy problem into a clear picture. Each branch shows one outcome and the chance of it. The branches leaving any one point always add up to one, because something must happen. To find the chance of a whole path, multiply the chances along its branches. If an outcome can be reached by more than one path, add the path chances together. And for a draw made without replacement, the second set of branches changes, because what is left has changed.

Pick the true facts about probability

Select the TWO statements that are true.

• The chances on the branches from one point add up to one
• For independent events you multiply along the branches
• Every probability is greater than one
• Mutually exclusive events can happen at the same time

Order how to use a tree diagram

Put the steps of using a tree diagram for combined events in order.

• Draw a branch for each outcome of the first event
• Write the chance on each branch
• Add branches for the second event
• Multiply along the branches for each full path
• Add the paths that give the outcome you want

Two counters from a bag

Follow how the chance changes when you take two counters from a bag without putting the first back. The bag: it holds three red counters and two blue, five in all.\n\nThe first draw: the chance of red is three in five, because three of the five counters are red. Why it changes: if a red is taken and kept, only four counters remain and two of them are red, so the chance of a second red is now two in four, which is one half. Along the branches: to find the chance of two reds, multiply the chance of the first red by the new chance of the second. Because the first draw changes what is left, the two draws are linked, not separate.

Complete the probability facts

A measure of how likely an event is, from impossible to certain, is its _____. Two events that cannot both happen at once are mutually _____. Events where one does not affect the other are _____. The chance of an event given that another has already happened is _____ probability.

Work out how many study neither

In a class of 30 students, 18 study French and 14 study German. Of these, 7 study both languages. Work out how many study neither language. First find how many study at least one, then subtract that from the total.

Build a probability point

Choose the words that complete this statement about combining events.

Make the probability choice

Read each situation and choose the correct method, then think about why.

• You want the chance of getting heads on two separate coin tosses. What do you do?
• You want the chance of rolling a two or a five on one die. What do you do?
• You draw two counters from a bag without replacing the first. Are the draws independent?

Explain how to work out probabilities

A classmate finds probability confusing. Explain how to work out probabilities using what this module has covered.

• Explain the probability scale from impossible to certain
• Explain the difference between mutually exclusive and independent events
• Explain how to use a tree diagram for combined events
• Explain what conditional probability means
• Finish by explaining how a Venn diagram helps sort outcomes