Elementary probability

Unit 8 Elementary probability

Key unit competence: Calculate theoretical and experimental probabilities to evaluate the likelihood of events in everyday situations.

Key ideas

• An experiment (trial) has outcomes; the set of all possible outcomes is the sample space $S$. An event is a subset of the sample space.
• Types of events: impossible (probability 0), certain (probability 1), equally likely, mutually exclusive (cannot happen together), complementary ($A$ and "not $A$").
• Theoretical probability for equally likely outcomes: $P(A)=\frac{n(A)}{n(S)}$.
• Experimental probability (relative frequency) $=\frac{\text{number of times the event happened}}{\text{number of trials}}$. As the number of trials increases, it gets closer to the theoretical probability.
• Basic rules: $0\le P(A)\le 1$; $P(\text{not }A)=1-P(A)$; the probabilities of all outcomes add up to 1; for mutually exclusive events $P(A\text{ or }B)=P(A)+P(B)$.
• Expected number of times an event occurs in $n$ trials $=n\times P(A)$.

Formulas to know

Common misconceptions

• "After 5 heads, a tail is more likely." No: each toss of a fair coin is independent; $P(\text{tail})$ is still $\frac{1}{2}$.
• "A probability can be $\frac{3}{2}$." No: every probability lies between 0 and 1.
• "There are two outcomes (win or lose), so $P(\text{win})=\frac{1}{2}$." Only if the outcomes are equally likely.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. List the sample space when a die is thrown and find the probability of a number less than 3.

   Show answer

   $S=\{1,2,3,4,5,6\}$; favourable $\{1,2\}$; $P=\frac{2}{6}=\frac{1}{3}$.

   Common mistake: Including 3 in "less than 3".

2. In a raffle, 500 tickets are sold. Gasana buys 20. What is the probability he wins the single prize? And that he does not?

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   $P(\text{win})=\frac{20}{500}=\frac{1}{25}$; $P(\text{not})=\frac{24}{25}$.

   Common mistake: Answering $\frac{1}{20}$.

3. A bag contains 4 red, 6 green and 2 yellow sweets. Find P(green), P(not red) and P(purple).

   Show answer

   $P(G)=\frac{6}{12}=\frac{1}{2}$; $P(\text{not }R)=\frac{8}{12}=\frac{2}{3}$; $P(\text{purple})=0$.

   Common mistake: Forgetting that an impossible event has probability 0.

4. Are "getting a 2" and "getting an even number" on one die mutually exclusive? Explain.

   Show answer

   No: 2 is even, so both can happen on the same throw.

   Common mistake: Confusing "different events" with "mutually exclusive events".

5. A coin was tossed 20, 100 and 1 000 times, with 13, 46 and 507 heads. Find the relative frequencies and comment.

   Show answer

   0.65, 0.46, 0.507. With more tosses the result approaches 0.5.

   Common mistake: Expecting exactly half heads in a small number of tosses.

6. Explain the difference between experimental and theoretical probability, with an example.

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   Theoretical probability comes from reasoning about equally likely outcomes (fair die: $P(6)=\frac{1}{6}$). Experimental probability comes from repeating the experiment (6 appears 18 times in 100 throws: 0.18).

   Common mistake: Thinking the two must always be equal.

7. The probability that a bus is late is 0.15. On how many of 40 school days do you expect it to be late?

   Show answer

   $40\times 0.15=6$ days.

   Common mistake: Answering 15 (treating 0.15 as a count).

8. A number is chosen from 1 to 30. Find the probability that it is a multiple of 4.

   Show answer

   Multiples: 4, 8, 12, 16, 20, 24, 28: $\frac{7}{30}$.

   Common mistake: Missing the last multiple 28.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. For mutually exclusive events $A$ and $B$, $P(A\text{ or }B)=$	a. $\frac{1}{2}$
   2. A bag has 5 red balls and $x$ blue balls. $P(\text{red})=\frac{1}{3}$. Find $x$.	b. 10
   3. Two dice are thrown. Find $P(\text{sum}=7)$.	c. $\frac{1}{6}$
   4. A number is chosen from 1 to 20. Find $P(\text{multiple of 3 or of 4})$.	d. $P(A)+P(B)$

   Show answer

   1 → d ($P(A)+P(B)$) · 2 → b (10) · 3 → c ($\frac{1}{6}$) · 4 → a ($\frac{1}{2}$)

   • They cannot happen together, so their probabilities add.
   • Step 1. $\frac{5}{5+x}=\frac{1}{3}$.
     Step 2. $5+x=15$, so $x=10$.
   • Step 1. There are $6\times 6=36$ outcomes. Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
     Step 2. $P=\frac{6}{36}=\frac{1}{6}$.
   • Step 1. Multiples of 3: 6 numbers; of 4: 5 numbers; of both (12): 1 number.
     Step 2. $6+5-1=10$, so $P=\frac{10}{20}=\frac{1}{2}$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $\frac{1}{2}$  ·  $\frac{12}{18}$  ·  mutually exclusive  ·  $\frac{2}{5}$
   1. A class has 18 boys and 12 girls. A pupil is chosen at random. The probability of choosing a girl is .
   2. Getting an even number and getting an odd number on one throw of a die are .
   3. A spinner has 4 equal sectors: red, red, green, yellow. Find $P(\text{red})$. →
   Show answer

   1: $\frac{2}{5}$ · 2: mutually exclusive · 3: $\frac{1}{2}$

   • $\frac{12}{30}=\frac{2}{5}$.
   • A number cannot be both even and odd.
   • 2 of the 4 equal sectors are red.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. A coin is tossed 50 times and shows heads 28 times. What is the experimental probability of heads?	a. 10
   2. A die is thrown 60 times. How many sixes do you expect?	b. $\frac{2}{5}$
   3. Two coins are tossed. Find $P(\text{exactly one head})$.	c. $\frac{1}{2}$
   4. A number is chosen from 1 to 10. Find $P(\text{prime})$.	d. 0.56

   Show answer

   1 → d (0.56) · 2 → a (10) · 3 → c ($\frac{1}{2}$) · 4 → b ($\frac{2}{5}$)

   • $\frac{28}{50}=0.56$.
   • $60\times \frac{1}{6}=10$.
   • Step 1. Sample space: HH, HT, TH, TT (4 equally likely outcomes).
     Step 2. Exactly one head: HT, TH. $P=\frac{2}{4}=\frac{1}{2}$.
   • Primes: 2, 3, 5, 7: $\frac{4}{10}=\frac{2}{5}$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   80  ·  50  ·  $\frac{1}{6}$  ·  $\frac{3}{5}$
   1. A fair die is thrown. Find $P(5)$. →
   2. A box has 10 pens: 4 are faulty. One pen is picked. Find $P(\text{good pen})$. →
   3. A spinner shows red with probability 0.35 and blue with 0.4; the rest is green. In 200 spins, how many greens do you expect? →
   Show answer

   1: $\frac{1}{6}$ · 2: $\frac{3}{5}$ · 3: 50

   • One favourable outcome out of 6.
   • 6 good pens: $\frac{6}{10}=\frac{3}{5}$.
   • Step 1. $P(\text{green})=1-0.35-0.4=0.25$.
     Step 2. Expected number $=0.25\times 200=50$.

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