Probability
Section: Statistics | Syllabus: Cambridge Primary Mathematics (0845)
The Language of Probability
Probability describes how likely an event is to happen, using words that range from impossible to certain. The same language can also compare and describe proportions within a group.
- Probability words range from impossible (will never happen) through unlikely, even chance, likely, to certain (will definitely happen)
- Comparing outcomes means judging which is more or less likely, e.g. picking a red counter is more likely than picking a blue one if there are more red counters
- The same language describes proportion too - what fraction or percentage of a group has a certain property
Probability words describe positions along a scale from impossible (0) to certain (1)
Worked Example: Describing and Comparing Outcomes
- Question: A bag contains 7 red counters and 3 blue counters. Describe the likelihood of picking a red counter, and compare it to picking a blue counter.
- Step 1: There are more red counters (7) than blue counters (3)
- Step 2: Picking red is more likely than picking blue, since there are more red counters to choose from
- Answer: Picking a red counter is likely; it is more likely than picking a blue counter
Common Mistakes
MistakeAssuming an event described as "unlikely" is impossible
Fixunlikely means it probably will not happen, but it still could - only "impossible" means it definitely cannot happen
Mutually Exclusive Events
Some events can happen at the same time as each other, while others cannot. Events that cannot happen at the same time are called mutually exclusive.
- Two events can happen at the same time if there is overlap between them, e.g. "rolling an even number" and "rolling a number greater than 3" on a dice can both be true for a 4 or a 6
- Two events are mutually exclusive if they cannot both happen at the same time, e.g. "rolling a 2" and "rolling a 5" on a single dice roll - only one result happens each time
- To check, ask: "could both of these happen from the very same outcome?"
Worked Example: Identifying Mutually Exclusive Events
- Question: A spinner lands on a number from 1 to 6. Are the events "landing on an even number" and "landing on a 3" mutually exclusive?
- Step 1: Check whether both could happen from the same spin - the spinner can only land on one number
- Step 2: 3 is odd, not even, so the two events describe different, non-overlapping outcomes
- Answer: Yes, these events are mutually exclusive - the spinner cannot land on an even number and on 3 at the same time
Common Mistakes
MistakeThinking any two different events are automatically mutually exclusive
Fixcheck for overlap first - events like "rolling an even number" and "rolling a number greater than 3" are NOT mutually exclusive, since 4 and 6 satisfy both
Chance Experiments and Trials
Some probabilities are easy to calculate just by counting possible outcomes, such as rolling a dice. Others can only be estimated by carrying out an experiment many times and observing what happens - and the more trials carried out, the more reliable the estimate becomes.
- A trial is one single attempt in a chance experiment, such as one roll of a dice or one spin of a spinner
- Some probabilities - like whether a drawing pin lands point-up or point-down - cannot be worked out by counting outcomes, and can only be estimated by running many trials and recording the results
- A small number of trials can give a misleading result; a large number of trials generally gives a more reliable estimate of the true probability
- The frequency of an outcome is how many times it occurred out of the total number of trials
With more trials, the proportion of outcomes settles closer to the true probability
Worked Example: Analysing the Results of an Experiment
- Question: A drawing pin was dropped 50 times. It landed point-up 32 times and point-down 18 times. Estimate the probability that the pin lands point-up.
- Step 1: Find the frequency of landing point-up out of the total trials: 32 out of 50
- Step 2: Write this as a fraction or decimal: 32/50 = 0.64
- Answer: The estimated probability of landing point-up is 32/50, or 0.64
Common Mistakes
MistakeAssuming a small number of trials gives an accurate probability
Fixa small number of trials can easily be misleading by chance - running more trials generally gives a more reliable estimate
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