Strand 1 — Statistics & Probability

Probability Rules

5th Year · 6th Year (Leaving Cert)

✓By the end of this lesson students will be able to apply the addition and multiplication rules for probability to solve problems involving independent, dependent, and mutually exclusive events.
✓By the end of this lesson students will be able to calculate conditional probabilities using appropriate formulae.
✓By the end of this lesson students will be able to understand and apply the concept of Bernoulli trials and the binomial distribution to solve problems (HL).
✓By the end of this lesson students will be able to calculate the expected value of a discrete random variable.

Key concepts

Addition Rule for Probability

The addition rule is used to find the probability of event A OR event B occurring. If events A and B are mutually exclusive (cannot happen at the same time), the probability of A or B is the sum of their individual probabilities. If they are not mutually exclusive (they can happen at the same time), the probability of their intersection must be subtracted to avoid double-counting.

P(A U B) = P(A) + P(B) (for mutually exclusive events) P(A U B) = P(A) + P(B) - P(A ∩ B) (for general events)

Multiplication Rule for Probability

The multiplication rule is used to find the probability of event A AND event B occurring. If events A and B are independent (the occurrence of one does not affect the probability of the other), the probability of A and B is the product of their individual probabilities. If they are dependent, the probability of A and B is the probability of A multiplied by the conditional probability of B given A.

P(A ∩ B) = P(A) * P(B) (for independent events) P(A ∩ B) = P(A) * P(B|A) (for dependent events)

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which means 'the probability of A given B'.

P(A|B) = P(A ∩ B) / P(B) (where P(B) > 0)

Bernoulli Trials and Binomial Distribution (HL)

A Bernoulli trial is an experiment with only two possible outcomes: 'success' or 'failure'. The probability of success (p) is constant for each trial, and the trials are independent. A sequence of n independent Bernoulli trials is called a binomial experiment. The binomial distribution models the probability of obtaining exactly k successes in n trials.

P(X=k) = (nCk) * p^k * (1-p)^(n-k) where: n = number of trials k = number of successes p = probability of success in a single trial (1-p) = probability of failure in a single trial nCk = n! / (k!(n-k)!) is the binomial coefficient, representing the number of ways to choose k successes from n trials.

Expected Value

The expected value (or expectation) of a discrete random variable is the long-run average value of the variable if the experiment is repeated many times. It is calculated by summing the product of each possible value of the variable and its corresponding probability.

E(X) = Σ x * P(X=x) For a binomial distribution, the expected value is simply E(X) = np.

Key facts to remember

1The complement rule states that P(A') = 1 - P(A), where A' is the event that A does not occur.
2The probability of any event E must be between 0 and 1, inclusive: 0 ≤ P(E) ≤ 1.
3Mutually exclusive events cannot occur at the same time, meaning their intersection is empty: P(A ∩ B) = 0.
4Independent events are those where the occurrence of one does not affect the probability of the other. For independent events, P(A|B) = P(A) and P(B|A) = P(B).
5Conditional probability P(A|B) is the probability of event A happening, given that event B has already happened.
6Bernoulli trials require a fixed number of independent trials, each with only two outcomes (success/failure), and a constant probability of success.
7The expected value represents the average outcome if an experiment is repeated many times.
8The sum of probabilities for all possible outcomes in a sample space must equal 1.

Worked examples

Example 1

A card is drawn at random from a standard deck of 52 playing cards. What is the probability that the card is a King or a Heart?

ILet K be the event that the card drawn is a King.

IILet H be the event that the card drawn is a Heart.

IIIThere are 4 Kings in a deck of 52 cards, so P(K) = 4/52 = 1/13.

IVThere are 13 Hearts in a deck of 52 cards, so P(H) = 13/52 = 1/4.

VThe events K and H are not mutually exclusive because there is one card that is both a King and a Heart (the King of Hearts).

VIP(K ∩ H) = P(King of Hearts) = 1/52.

VIIUsing the addition rule for non-mutually exclusive events: P(K U H) = P(K) + P(H) - P(K ∩ H).

VIIIP(K U H) = 1/13 + 1/4 - 1/52.

9To add these fractions, find a common denominator, which is 52:

10P(K U H) = (4/52) + (13/52) - (1/52).

11P(K U H) = (4 + 13 - 1) / 52 = 16/52.

12Simplify the fraction: 16/52 = 4/13.

Answer

The probability that the card is a King or a Heart is 4/13.

Remember to subtract the intersection when events are not mutually exclusive to avoid double-counting.

Example 2

In a class, 60% of students study Maths, 40% study Physics, and 20% study both. What is the probability that a student chosen at random studies Physics given that they study Maths?

ILet M be the event that a student studies Maths.

IILet P be the event that a student studies Physics.

IIIGiven probabilities:

IVP(M) = 0.60

VP(P) = 0.40

VIP(M ∩ P) = 0.20 (probability of studying both Maths AND Physics).

VIIWe need to find P(P|M), the probability of studying Physics given that they study Maths.

VIIIUsing the conditional probability formula: P(P|M) = P(P ∩ M) / P(M).

9Substitute the given values:

10P(P|M) = 0.20 / 0.60.

11P(P|M) = 1/3.

Answer

The probability that a student studies Physics given that they study Maths is 1/3.

The order matters in conditional probability: P(P|M) is not necessarily equal to P(M|P).

Example 3

A biased coin has a probability of landing heads of 0.6. The coin is tossed 4 times. (a) What is the probability of getting exactly 3 heads? (b) What is the expected number of heads?

IThis is a Bernoulli trial scenario, as there are a fixed number of trials (n=4), two outcomes (Heads/Tails), a constant probability of success (p=0.6 for Heads), and independent trials.

II(a) Probability of exactly 3 heads:

IIIIdentify parameters: n = 4 (number of tosses), p = 0.6 (probability of heads), k = 3 (number of heads).

IVThe probability of failure (tails) is 1-p = 1-0.6 = 0.4.

VUse the binomial probability formula: P(X=k) = (nCk) * p^k * (1-p)^(n-k).

VIP(X=3) = (4C3) * (0.6)^3 * (0.4)^(4-3).

VIICalculate 4C3: 4C3 = 4! / (3!(4-3)!) = 4! / (3!1!) = (4*3*2*1) / ((3*2*1)*(1)) = 4.

VIIIP(X=3) = 4 * (0.6)^3 * (0.4)^1.

9P(X=3) = 4 * (0.216) * (0.4).

10P(X=3) = 4 * 0.0864.

11P(X=3) = 0.3456.

12(b) Expected number of heads:

13For a binomial distribution, the expected value E(X) = np.

14E(X) = 4 * 0.6.

15E(X) = 2.4.

Answer

(a) The probability of getting exactly 3 heads is 0.3456. (b) The expected number of heads is 2.4.

The expected value does not have to be an integer, as it represents a long-run average.

Common mistakes

✗Confusing mutually exclusive events with independent events. Mutually exclusive events cannot happen together (P(A∩B)=0), while independent events can, but one doesn't influence the other (P(A∩B)=P(A)P(B)).
✗Incorrectly applying the addition rule by forgetting to subtract P(A ∩ B) when events are not mutually exclusive.
✗Incorrectly applying the multiplication rule by using P(A)P(B) for dependent events instead of P(A)P(B|A).
✗Misinterpreting the 'given' condition in conditional probability, often confusing P(A|B) with P(B|A) or P(A ∩ B).
✗Errors in calculating the binomial coefficient (nCk) or powers in the binomial probability formula.

Exam tips

★Always clearly define your events using capital letters (e.g., A = 'event of drawing an ace') to avoid confusion.
★Draw Venn diagrams for addition rule problems and tree diagrams for sequential events or conditional probability problems to visualise the relationships between events.
★Before applying any formula, identify whether events are mutually exclusive/non-mutually exclusive or independent/dependent. This dictates which version of the rule to use.
★For Bernoulli trials and binomial distribution questions, clearly state the values of n (number of trials), p (probability of success), and k (number of successes) before starting calculations.

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