Functional Skills: Probability

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Functional Skills: Probability Revision

Probability

Probability is a measurement of the likelihood of an event happening. We can use fractions, decimals and percentages to describe probabilities.

There are 4 skills you need to understand for probability.

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Skill 1: The Probability Scale

The probability scale summarises the likelihood of an event occurring:

  • At the lower end of the scale, the likelihood of an event occurring is very small, so the probability is close to zero.
  • As we move up the scale, events become increasingly likely.
  • When the probability is \dfrac{1}{2}, there is an equal chance of an event occurring/not occurring.
  • If the probability is less than \dfrac{1}{2}, the event is unlikely to occur.
  • If the probability is greater than \dfrac{1}{2}, the event is likely to occur.
  • Events that are impossible have a probability of 0 and events that are certain have a probability of 1
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Notes:

  • We usually express probabilities as fractions or decimals, but we can also use percentages. For example, a probability of \dfrac{1}{2} means there is a 50\% chance of the event occurring and a probability of \dfrac{3}{4} means there is a 75\% chance of an event occurring etc.
  • You may sometimes see notation such as P(\text{event}) which means the probability of an event happening – the word in the bracket represents what the event is, e.g. you may see P(\text{odd}) which means the probability of something being odd, or P(\text{red}) which means the probability of something being red.

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Skill 2: Calculating Probabilities

We can calculate the probability of an event occurring if we know the total number number of possible outcomes.

Example: If we want to calculate the probability of rolling an even number on a standard die, we need the total number of possible outcomes. There are \textcolor{blue}{6} numbers on a standard die, of which \textcolor{red}{3} are even (2, 4 and 6). We can calculate the probability as follows:

\begin{aligned}\text{Probability}&=\dfrac{\color{red}{\text{Number of ways an event can happen}}}{\color{blue}{\text{Total number of possible outcomes}}} \\[1.5em] &= \dfrac{\color{red}{3}}{\color{blue}{6}} = \dfrac{\color{red}{1}}{\color{blue}{2}} = 50\% = 0.5\end{aligned}
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Skill 3: Probabilities Add up to 1

All probabilities must add up to 1 (as decimals or fractions). So, we can calculate the probability of an event not happening by subtracting the probability that it will happen from 1:

Probability of event not happening = \, 1 \, - probability of event happening

 

Example: The probability of it raining tomorrow is \textcolor{red}{0.4}. What is the probability of it not raining tomorrow?

\begin{aligned} \text{Probability of no rain tomorrow} &= 1  - \text{probability of rain tomorrow} \\ &= 1 - \textcolor{red}{0.4} = \textcolor{blue}{0.6} \end{aligned}

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Skill 4: Events Happening Multiple Times

To find the probability of an event happening multiple times, we multiply the probabilities together.

Example: Mehmed has a coin. The probability of landing on tails is \textcolor{red}{0.5}.

Mehmed flips the coin three times. What is the probability of landing on tails on all three occasions?

\text{Probability of three tails} = \textcolor{red}{0.5} \times \textcolor{red}{0.5} \times \textcolor{red}{0.5} = \textcolor{blue}{0.125}

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Skill 5: Probability Tree Diagrams

The probabilities of multiple events occurring can be represented on a probability tree diagram, as shown on the right.

The probability for each outcome is shown on each corresponding branch.

Example: Beth needs to catch two trains to get to work.

The probability that the first train is on time is 0.9 and the probability that the second train is on time is 0.8

Whether or not the first train is on time or late does not affect the probability that the second train is on time or late.

Complete the probability tree diagram and calculate the probability that both trains are on time.

Step 1: Complete the probability tree diagram.

As stated in Skill 3, probabilities of all outcomes in a single event must add up to \bf{\textcolor{#f95d27}{1}}

Therefore, the probability that the first train is late is

1-0.9=0.1

 

Similarly, the probability that the second train is late is

1-0.8=0.2

 

Step 2: Calculate the probability of both trains being on time.

As stated in Skill 4, to find the probability of multiple events occurring we multiply the probabilities together.

So in this case, we multiply the probability of the first train being on time by the probability of the second train being on time:

0.9\times0.8=0.72
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Example: Probability Scale

Match up the following events with the labels on the probability scale.

i) The probability that it rains in any given week in winter

ii) The probability of getting ‘tails’ on a coin toss

iii) The probability that you will have a birthday this year

iv) The probability that you win the lottery

[4 marks]

i) In any given week in winter, it is fairly likely (but not certain) that it will rain. The probability that it rains in any given week in winter will therefore be between \dfrac{1}{2} and 1, which matches up to \color{black}C

ii) The probability of getting tails on a coin toss is \dfrac{1}{2}, as there is an equal chance of getting heads or tails. This matches up to \color{black}B on the probability scale

iii) It is certain you will have a birthday each year (unless you are born on the 29th February in a leap year!), so the probability is 1, which matches up to \color{black}D

iv) The likelihood that you win the lottery is extremely small (but not impossible) so it will be very close to 0. This matches up to \color{black}{A}

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Functional Skills: Probability Example Questions

\begin{aligned} \text{Probability} &= \dfrac{\text{number of ways an outcome can happen}}{\text{total number of possible outcomes}} \\ &= \dfrac{20}{250} = 0.08 = 8\% \end{aligned}

Probability of blue counter = 1 \, - probability of red counter = 1 - 0.35 = 0.65

The probability of rolling the a number 3 is \dfrac{1}{6}, since a die has 6 sides.

Therefore, to find the probability of rolling a number 3 on all three occasions, we multiply the probability of rolling a 3 by itself and then by itself again:

Probability = \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{1}{6} = \dfrac{1}{216}

a) The probability that Lois doesn’t win a prize is

1-0.4=0.6

 

Here is the completed probability tree diagram:

b) The probability that Lois doesn’t win a prize on either attempts is

0.6\times0.6=0.36

Additional Resources

PFS

Exam Tips Cheat Sheet

FS Level 2
PFS

Formula Booklet

FS Level 2

Functional Skills: Probability Worksheet and Example Questions

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Probability L1

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Probability L2

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