Probability is a type of measurement used to express the likelihood of an event occurring. Just as we measure length or area, probability measures how likely or confident we are that a particular event will happen.

Probability is measured on a scale from 0 to 1:

• A probability of 0 means the event is impossible — it cannot occur (e.g., rolling a 7 on a standard 6-sided die).

• A probability of 1 means the event is certain — it will definitely occur (e.g., choosing a red sweet from a bag containing only red sweets).

Most events have probabilities strictly between 0 and 1, expressing varying degrees of likelihood.

Randomness refers to a situation or action where you cannot predict exactly what will happen, even though you may know all the possible outcomes.

Examples:

1. Tossing a coin: You know the outcome will be either Heads (H) or Tails (T), but you cannot be sure which one will come up in a single toss.

2. Rolling a die: The possible outcomes are 1, 2, 3, 4, 5, or 6, but you do not know which number will appear on any particular roll.

The sample space, denoted by S, is the set of all possible outcomes of a random experiment. Each outcome is an element of the sample space, and no outcome should appear more than once. The number of elements is called the sample size, denoted n(S).

(i) Tossing a single coin:

(ii) Rolling a standard 6-sided die:

Experimental Probability: Based on actual data collected from trials or experiments. It is calculated as:

It can differ from theoretical probability, especially when the number of trials is small.

Theoretical Probability: Based on reasoning, assuming all outcomes are equally likely. No experiment is needed. It is calculated as:

Gambler’s Fallacy is the mistaken belief that if a random event has occurred many times in a row, the opposite outcome is “due” to happen next.

Example: If a fair coin shows Heads six times in a row, many people believe Tails is more likely on the next flip. However, this is incorrect.

The coin has no memory of past flips. Every toss is an independent event. The probability of getting Tails on any flip is always:

Sample Space: S = {1, 2, 3, 4, 5, 6}, so n(S) = 6

(i) Getting the number 4:

Favourable outcomes = {4}, so favourable count = 1

(ii) Getting an even number:

Favourable outcomes = {2, 4, 6}, so favourable count = 3

The word PROBABILITY has 11 letters: P-R-O-B-A-B-I-L-I-T-Y

Number of all possible outcomes = 11

Number of B’s in the word = 2 (favourable outcomes)

Experimental probability:

Theoretical probability:

The experimental probability (0.16) is close to but not exactly equal to the theoretical probability (≈ 0.167). This difference is expected when the number of trials is small. As the number of trials increases, the two values get closer — this is the Law of Large Numbers.

Sample Space: S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, so n(S) = 10

Even numbers in the sample space: {2, 4, 6, 8, 10} → 5 favourable outcomes

Total balls = 3 + 2 + 1 = 6, so n(S) = 6

Balls that are NOT red = blue + green = 2 + 1 = 3

When two coins are tossed, each coin can show H or T independently.

Event E: “At least one head appears” means one or both coins show Heads.

Probability: P(E) = 3/4 = 0.75 or 75%

Relative frequency (experimental probability) is calculated from observed data:

Students who like football = 15, Total students = 50

(i) The next Monday comes after Sunday:

This is Certain (probability = 1). The order of days of the week is fixed and never changes. Monday always follows Sunday.

(ii) It will snow in Mumbai in July:

This is Impossible (probability ≈ 0). Mumbai has a tropical climate and is located near the equator. July is during the hot monsoon season — snowfall is essentially impossible there.

An event is any single possible result or a combination of results that might happen when a random experiment is performed. An event is a subset of the sample space.

Example — Rolling a 6-sided die:

Sample Space: S = {1, 2, 3, 4, 5, 6}

Event E: “The number rolled is greater than 4”

Total students surveyed = 50, Students preferring mango = 20

To estimate for the whole school (1500 students), we apply the probability:

The two main ways of objectively measuring the probability of an event are:

1. Experimental Probability (Evidence from experience): This involves collecting data by performing an experiment multiple times, or by analysing statistical data from past observations. The probability is estimated by computing the relative frequency.

   P(E) = (Number of times the event occurred) ÷ (Total number of trials)

   Example: Rolling a die 50 times and getting a 4 exactly 8 times → P(4) = 8/50 = 0.16 or 16%.

   Most useful when dealing with real-world, complex events where theoretical methods cannot be used.

2. Theoretical Probability: This assumes all possible outcomes are equally likely (a “perfectly fair” situation). No experiment or data is needed.

   P(A) = (Number of favourable outcomes) ÷ (Number of possible outcomes)

   Example: Rolling a fair die → P(getting a 4) = 1/6 ≈ 0.167 or 16.7%.

   Most useful when outcomes are symmetric and equally likely (coins, dice, cards).

3. Statistical Probability (Special case of experimental): Uses data collected from a sample of a population to estimate probabilities for the whole population.

   Example: If 20 out of 50 surveyed students like mango, we estimate 40% of all students like mango.

4. Key difference: Experimental probability can vary from the theoretical value, especially with few trials. As trials increase, the experimental value converges to the theoretical value — this is the Law of Large Numbers.

Given: Sample of 30 sweets. Red = 10, Green = 8, Yellow = 7, Blue = 5. Total sweets in bag = 600.

1. Find P(green):

   P(green) = 8/30 = 4/15 ≈ 0.267 or 26.7%

2. Find P(yellow):

   P(yellow) = 7/30 ≈ 0.233 or 23.3%

3. Estimate yellow sweets in 600:
   Expected yellow = (7/30) × 600 = 140 sweets
4. Estimate red sweets in 600:
   Expected red = (10/30) × 600 = 200 sweets
5. Estimate green sweets in 600:
   Expected green = (8/30) × 600 = 160 sweets
6. Estimate blue sweets in 600:
   Expected blue = (5/30) × 600 = 100 sweets

A tree diagram is a visual representation used to list all possible outcomes of a multi-step experiment (where a series of independent trials takes place). Each branch represents one outcome, and branches split to show paths for subsequent events. It is useful for: (a) listing all outcomes in a sample space, and (b) calculating probabilities of combined events.

Experiment: Tossing a fair coin two times.

Tree structure:
Start → First Toss: H → Second Toss: H (outcome: HH) or T (outcome: HT)
Start → First Toss: T → Second Toss: H (outcome: TH) or T (outcome: TT)

Each outcome has equal probability = 1/4

1. P(HH) — Both coins show Heads:

   Favourable outcomes: {HH} → 1 outcome

   P(HH) = 1/4 = 0.25 or 25%

2. P(exactly one Head) — One coin shows H, the other T:

   Favourable outcomes: {HT, TH} → 2 outcomes

   P(exactly one Head) = 2/4 = 1/2 = 0.5 or 50%

3. P(at least one Tail) — One or both coins show Tails:

   Favourable outcomes: {HT, TH, TT} → 3 outcomes

   P(at least one Tail) = 3/4 = 0.75 or 75%

Given: Sample = 40 students. Science = 14, Arts = 11, Sports = 9, Debate = 6. Total students in school = 800.

1. P(Arts Club):

   P(Arts) = 11/40 = 0.275 or 27.5%

2. P(Sports Club):

   P(Sports) = 9/40 = 0.225 or 22.5%

3. Estimate Sports Club students in school:

   Expected = (9/40) × 800 = 180 students

4. Find P for each club to identify most likely:

   P(Science) = 14/40 = 0.35 (35%) — highest

   P(Arts) = 11/40 = 0.275 (27.5%)

   P(Sports) = 9/40 = 0.225 (22.5%)

   P(Debate) = 6/40 = 0.15 (15%)

5. Conclusion: Since P(Science) is highest (35%), a new random student is most likely to prefer Science Club.

Given: Total cases = 1000 (this is experimental/statistical probability).

1. Verify total: 20 + 210 + 325 + 445 = 1000 ✓
2. P(less than 4000 km):

   P = 20/1000 = 1/50 = 0.02 or 2%

3. Cases between 4001 and 14000 km (combining two categories):

   4001–9000: 210 cases; 9001–14000: 325 cases

   Total in this range = 210 + 325 = 535 cases

4. P(between 4001 and 14000 km):

   P = 535/1000 = 0.535 or 53.5%

5. P(more than 14000 km):

   P = 445/1000 = 0.445 or 44.5%

6. Verify: 0.02 + 0.535 + 0.445 = 1.0 ✓ (probabilities of all outcomes sum to 1)

The word PEACE has 5 letters: P, E, A, C, E. So n(S) = 5.

1. Identify letter frequencies:

   P appears 1 time; E appears 2 times; A appears 1 time; C appears 1 time.

2. P(drawing a P, E, or C):

   Favourable cards: P (1) + E (2) + C (1) = 4

   P(P, E, or C) = 4/5 = 0.8 or 80%

3. P(not drawing an E):

   Cards that are NOT E: P (1) + A (1) + C (1) = 3

   P(not E) = 3/5 = 0.6 or 60%

4. P(drawing a vowel):

   Vowels in PEACE: E, A, E → 3 vowel cards

   P(vowel) = 3/5 = 0.6 or 60%

1. Sample space for two coins:

   Each coin shows H or T. Combining all possibilities:

   S₂ = {HH, HT, TH, TT}   n(S₂) = 4

2. Sample space for three coins:

   We extend by adding H or T for the third coin to each outcome of two coins:

   S₃ = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}   n(S₃) = 8

3. Why different sizes? Each additional coin doubles the sample size (since each coin has 2 outcomes). For 2 coins: 2² = 4. For 3 coins: 2³ = 8.
4. Event E: “Exactly two Heads” from three coins:

   Favourable outcomes: {HHT, HTH, THH} → 3 outcomes

5. Calculate probability:

   P(exactly two heads) = 3/8 = 0.375 or 37.5%

Using a tree diagram approach: each snack branches into 2 drink options.

1. Branch 1 (Samosa): → Chai (Samosa, Chai)   → Lassi (Samosa, Lassi)
2. Branch 2 (Pakora): → Chai (Pakora, Chai)   → Lassi (Pakora, Lassi)
3. Branch 3 (Bhaji): → Chai (Bhaji, Chai)   → Lassi (Bhaji, Lassi)
4. Full sample space:

   S = {(Samosa,Chai), (Samosa,Lassi), (Pakora,Chai), (Pakora,Lassi), (Bhaji,Chai), (Bhaji,Lassi)}

   n(S) = 6

5. P(choosing Samosa): Outcomes with Samosa = {(Samosa,Chai), (Samosa,Lassi)} = 2

   P(Samosa) = 2/6 = 1/3 ≈ 0.333 or 33.3%

6. P(choosing Lassi): Outcomes with Lassi = {(Samosa,Lassi), (Pakora,Lassi), (Bhaji,Lassi)} = 3

   P(Lassi) = 3/6 = 1/2 = 0.5 or 50%

7. P(Pakora with Chai): Only {(Pakora,Chai)} = 1 outcome

   P(Pakora,Chai) = 1/6 ≈ 0.167 or 16.7%

Sample Space: S = {1, 2, 3, 4, 5, 6, 7, 8}, so n(S) = 8. All outcomes are equally likely.

1. P(pointing at 8): Only 1 favourable outcome: {8}

   P(8) = 1/8 = 0.125 or 12.5%

2. P(odd number): Odd numbers in 1–8: {1, 3, 5, 7} → 4 outcomes

   P(odd) = 4/8 = 1/2 = 0.5 or 50%

3. P(greater than 2): Numbers > 2: {3, 4, 5, 6, 7, 8} → 6 outcomes

   P(>2) = 6/8 = 3/4 = 0.75 or 75%

4. P(less than 9): All numbers 1–8 are less than 9 → 8 outcomes

   P(<9) = 8/8 = 1 (Certain event)

5. P(multiple of 3): Multiples of 3 in 1–8: {3, 6} → 2 outcomes

   P(multiple of 3) = 2/8 = 1/4 = 0.25 or 25%

1. Experimental probability of rolling a 3:

   Event occurred 3 times out of 12 trials

   P_exp(3) = 3/12 = 1/4 = 0.25 or 25%

2. Theoretical probability of rolling a 3:

   Only 1 face shows 3, out of 6 equally likely faces

   P_theo(3) = 1/6 ≈ 0.167 or 16.7%

3. Are they the same? No. Experimental P = 0.25, Theoretical P ≈ 0.167. They differ.
4. Why do they differ? With only 12 trials, the results are affected by chance variation. A small sample may not be representative of the true probability. Getting 3 exactly 3 times in 12 rolls happens partly by luck.
5. What happens with 6000 rolls? As the number of trials increases, the experimental probability converges towards the theoretical probability. With 6000 rolls, we would expect 3 to appear approximately 1000 times (1000/6000 = 1/6 ≈ 0.167). This principle is known as the Law of Large Numbers.