Probability • Topic 3 of 3

Probability Games and Applications

Probability Games

Probability games use chance to determine outcomes. Understanding probability helps players make better decisions and understand their chances of winning.

Common Probability Games:

Game	Probability Concept	Example
Dice games	Rolling specific numbers	Craps, Monopoly
Card games	Drawing specific cards	Poker, Blackjack
Spinner games	Spinning to land on sections	Board games, Wheel of Fortune
Lottery/raffle	Chance of winning	Raffle tickets
Coin toss games	Fairness of tosses	Starting a football match

Fair Games vs Unfair Games:

Fair game: Each player has equal probability of winning
Unfair game: One player has a higher probability of winning

Calculating Expected Value:

Expected value = (Probability of winning × Amount won) - (Probability of losing × Amount lost)

If expected value = 0, the game is fair.

If expected value > 0, the game favors the player.

If expected value < 0, the game favors the house/organizer.

Making Predictions Using Probability:

To predict how many times an event will occur:

Predicted occurrences = Probability × Number of trials

Example: If probability of rolling a 6 is $\frac{1}{6}$, in 600 rolls we predict:

$\frac{1}{6} \times 600 = 100$ sixes

Real-world Applications:

Weather forecasting: Probability of rain helps plan activities
Medical testing: Probability of accurate test results
Sports: Probability of winning based on team statistics
Insurance: Probability of accidents to set premiums
Gambling: Casinos use probability to ensure profit

Solved Examples

Worked Example

In a raffle, 200 tickets are sold. You buy 5 tickets. What is the probability that you win the first prize? If the first prize is ₹1000, what is your expected winning?

Solution

Step 1: P(win) = $\frac{\text{Your tickets}}{\text{Total tickets}} = \frac{5}{200} = \frac{1}{40} = 0.025$
Step 2: Expected winning = Probability × Prize amount
Step 3: Expected winning = $0.025 \times 1000 = ₹25$

Answer: Probability = $\frac{1}{40} = 0.025$ or 2.5%. Expected winning = ₹25.

Worked Example

A game uses a spinner divided into 4 equal sections labeled 1, 2, 3, 4. You win if you spin an odd number. If you spin 80 times, how many wins would you predict?

Solution

Step 1: P(odd) = P(1 or 3) = $\frac{2}{4} = \frac{1}{2} = 0.5$
Step 2: Predicted wins = Probability × Number of spins
Step 3: Predicted wins = $0.5 \times 80 = 40$ times

Answer: Predicted 40 wins out of 80 spins.

Worked Example

**Example 3 (Challenging):** Two friends play a game. Player A rolls a die. If the number is 1 or 2, Player A wins. If the number is 3, 4, or 5, Player B wins. If the number is 6, they roll again. Is this a fair game? What is the probability that Player A wins eventually? *Solution:* - Step 1: On

Solution

Step 1: On each roll (ignoring rerolls): P(A wins on this roll) = 2/6 = 1/3
Step 2: P(B wins on this roll) = 3/6 = 1/2
Step 3: P(reroll) = 1/6
Step 4: Probability A wins eventually = P(A wins on first roll) + P(reroll) × P(A wins eventually)
Step 5: Let x = P(A wins eventually). Then:
$x = \frac{1}{3} + \frac{1}{6}x$
Step 6: $x - \frac{1}{6}x = \frac{1}{3}$
Step 7: $\frac{5}{6}x = \frac{1}{3}$
Step 8: $x = \frac{1}{3} \times \frac{6}{5} = \frac{6}{15} = \frac{2}{5} = 0.4$
Step 9: P(B wins eventually) = $1 - 0.4 = 0.6$
Step 10: Since P(A) ≠ P(B), the game is NOT fair (favors Player B)

Answer: Game is not fair. P(A wins) = $\frac{2}{5} = 0.4$, P(B wins) = $\frac{3}{5} = 0.6$

Key Points

Prediction: Number of occurrences = Probability × Number of trials
Fair game: Each player has equal chance of winning (probability = 0.5)
Expected value helps determine if a game favors the player or the house
Probability can help make informed decisions in games and real life
Tree diagrams help visualize sequential events
More trials lead to more reliable predictions

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.A game is fair if all players have:

Explanation: Equal chance of winning.

Q2.Rolling two dice gives how many possible outcomes?

Explanation: $6\times6=36$.

Q3.Tossing two coins gives how many outcomes?

Explanation: HH, HT, TH, TT.

Q4.$P(\text{two heads when tossing two coins})=$

Explanation: $\tfrac14$.

Practice more MCQs → 📝 Assignment · 35 marks