Year 9: Probability of simple events

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1. What is probability?

Probability measures how likely an event is to occur. It is written as a number between 0 impossible and 1 certain. You can also express probability as a fraction, decimal or percentage.

2. The Probability scale

• 0 – Impossible for example rolling a 7 on a standard die
• 0.25 – Unlikely
• 0.5 – Even chance for example heads on a fair coin
• 0.75 – Likely
• 1 – Certain for example rolling a number less than or equal to 6 on a standard die

3. Calculating probability of simple events

$\mathrm{Pr}\left(E\right)=\text{favourable}\text{outcomes}∕\text{total}\text{outcomes}$

1. Coin toss: two possible outcomes H and T. $\mathrm{Pr}\left(\text{Heads}\right)=1∕2=0.5=50\text{percent}$
2. Six sided die: six outcomes 1 to 6. $\mathrm{Pr}\left(\text{rolling}a4\right)=1∕6$
3. Spinner with four equal colours: four outcomes. $\mathrm{Pr}\left(\text{landing}\text{on}\text{blue}\right)=1∕4$

4. Complementary events

The complement of event E written as not E is every outcome where E does not happen. They add to 1:

$\mathrm{Pr}\left(E\right)+\mathrm{Pr}\left(\text{not}E\right)=1$

Example: If $\mathrm{Pr}\left(\text{rain}\right)=0.3$ then $\mathrm{Pr}\left(\text{no}\text{rain}\right)=1\text{minus}0.3=0.7$.

5. Experimental versus theoretical probability

Theoretical probability is what you expect mathematically. Experimental probability is what you observe when you run a trial.

Experimental probability gets closer to theoretical probability as the number of trials increases. This idea is called the Law of Large Numbers.

6. Independent events

Two events are independent if the outcome of one does not affect the other. For independent events A and B:

$\mathrm{Pr}\left(A\text{and}B\right)=\mathrm{Pr}\left(A\right)\text{multiplied}by\mathrm{Pr}\left(B\right)$

Example: Tossing a coin and rolling a die. $\mathrm{Pr}\left(\text{Heads}\text{and}a6\right)=1∕2\text{multiplied}by1∕6=1∕12$.

7. Common mistakes

• Forgetting to count every possible outcome.
• Mixing different probability forms fractions versus percentages incorrectly.
• Assuming events are independent when they are not.

8. Practice questions

1. What is the probability of rolling an even number on a fair die?
2. A bag contains three red five blue and two green marbles. One marble is drawn at random. Find $\mathrm{Pr}\left(\text{green}\right)$.
3. A spinner is divided into eight equal sectors numbered 1 to 8. Find $\mathrm{Pr}\left(\text{number}\text{greater}\text{than}5\right)$.
4. A card is drawn from a standard 52 card deck. What is the probability it is a heart?
5. You toss two coins. What is the probability of getting exactly one head?

9. Solutions

1. Even numbers 2 4 6 gives three favourable outcomes. Probability 3∕6 = 1∕2.
2. Green marbles 2∕10 = 1∕5 = 0.2.
3. Numbers greater than 5 are 6 7 8. Probability 3∕8.
4. Hearts 13∕52 = 1∕4.
5. Possible outcomes HH HT TH TT. Exactly one head occurs in HT and TH. Probability 2∕4 = 1∕2.