Probability Notes I

Probability in Roulette

• Probability of landing on 5

  • There are 38 possible outcomes in a standard roulette game (numbers 1-36, 0, and 00).

  • Probability (P) that the ball lands on 5:

    • P(5) = Number of favorable outcomes / Total outcomes = 1/38.

• Probability of landing not on 5

  • P(not 5) = 1 - P(5) = 37/38.

Outcomes and Odds

• Probability of landing on multiples of 5:

  • Numbers that are multiples of 5 in roulette: 5, 10, 15, 20, 25, 30, 35 (7 favorable outcomes).

  • P(multiples of 5) = 7/38.

• Actual odds against landing on 5:

  • Odds against an event = P(not event) / P(event) = P(not 5) / P(5) = (37/38) / (1/38) = 37:1.

• Actual odds in favor of landing on 5:

  • Odds in favor of an event = P(event) / P(not event) = P(5) / P(not 5) = (1/38) / (37/38) = 1:37.

Die Rolls

• Probability of rolling a 2 or 3 on a fair die:

  • Outcomes: {1, 2, 3, 4, 5, 6} (6 total outcomes).

  • P(2 or 3) = 2/6 = 1/3.

• Probability of rolling a 2, 3, or 4:

  • P(2 or 3 or 4) = 3/6 = 1/2.

Coin Tosses

• Probability of getting Heads (H) twice in two tosses:

  • P(H,H) = P(H) * P(H) = (1/2)*(1/2) = 1/4.

• Probability of getting HHH in three tosses:

  • P(H,H,H) = (1/2)^3 = 1/8.

Card Probabilities

• Probability of picking a spade or a king from a deck:

  • Total spades = 13, Total Kings = 4 (but King of Spades counted twice).

  • P(spade or king) = (13 + 3) / 52 = 16/52 = 4/13.

• Probability of first pick being the king of diamonds:

  • P(King of Diamonds) = 1/52.

• Probability of drawing another king after King of Diamonds (without replacement):

  • P(King 2nd | King 1st not replaced) = 3/51.

• Joint probability of picking King of Diamonds and then another King:

  • P(King 1st and King 2nd) = (1/52)*(3/51) = 3/2652.

Probability of Defects

• Probability of getting at least one defective product from 12 purchases (defect rate 15%):

  • Using complement rule: P(at least one defective) = 1 - P(no defective).

  • P(no defective) = (0.85)^12, hence P(at least one defective) = 1 - (0.85)^12.

Dish Orders at a Restaurant

• Dish order ratios at a restaurant with 60% men and 40% women:

  • 80% of men order Dish A; therefore, men for A = 0.6 * 0.8 = 0.48.

  • 70% of women order Dish B; therefore, women for B = 0.4 * 0.7 = 0.28.

  • Ratio of orders for Dish A and B:

    • Total for A = 0.48 + (0.4 * 0.3) = 0.48 + 0.12 = 0.60 (from women).

    • Total for B = 0.28 + 0.12 = 0.40.

Probability of Scooters from Plants

• Scooter quality from two plants:

  • Plant I manufactures 80% of scooters, 85% of those are standard quality.

  • Plant II manufactures 20%, 65% of those are standard quality.

  • Applying Bayes' theorem:

    • P(Plant I | Standard Quality) = (P(Plant I) * P(Standard | Plant I)) / P(Standard).

    • P(Standard) = P(Plant I) * P(Standard | Plant I) + P(Plant II) * P(Standard | Plant II).

Cumulative Distribution Function (CDF)

• Definition and properties of CDF:

  • FX(x) = P(X ≤ x).

  • Properties: 0 ≤ FX(x) ≤ 1, FX(∞) = 1, FX(−∞) = 0, FX(x1) ≤ FX(x2) if x1 ≤ x2.

  • Probabilities can be derived as follows:

    • P(a ≤ X ≤ b) = FX(b) − FX(a).

• Example CDF calculation: Given FX(x):

  • P(X ≤ 1), P(1 ≤ X ≤ 2), and P(X > 1) are calculated from the defined regions in FX.

Probability Density Function (PDF)

• Definition of PDF: fX(x) is the derivative of FX.

• Properties of PDF: fX(x) ≥ 0, ∫ fX(x) dx over the entire space = 1.

• Example problem of determining constant k for given PDF: fX(x) = kx for 0 < x < 1.

Mean and Variance of Random Variables

• To find the mean or expected value of X given a PDF: E(X) = ∫ x * fX(x) dx.

• To find variance: Var(X) = E(X^2) - (E(X))^2, where E(X^2) = ∫ x^2 * fX(x) dx.

Probability in Roulette

Probability of Landing on a Specific Number

• Outcomes in Roulette: In a standard game of roulette, there are 38 possible outcomes. These include the numbers 1 through 36, as well as the green 0 and 00. This setup means players face a diverse range of potential landing spots for the ball.

• Calculating Probability: The probability (P) that the ball lands on a specific number, such as 5, can be quantified as follows:

  • P(5) = Number of favorable outcomes (landing on 5, which is 1) / Total outcomes (38) = 1/38.

Probability of Not Landing on a Specific Number

• To find the probability of the ball not landing on the number 5, we can derive it by:

  • P(not 5) = 1 - P(5) = 37/38. This demonstrates that there is a much higher likelihood of not hitting a specific number compared to hitting it.

Outcomes and Odds

• Multiple Outcomes: The numbers that are multiples of 5 within the roulette game include 5, 10, 15, 20, 25, 30, and 35. In total, there are 7 favorable outcomes that represent multiples of 5, giving:

  • P(multiples of 5) = 7/38.

• Calculating Odds: The actual odds against landing on 5 are expressed as:

  • Odds against an event = P(not event) / P(event) = P(not 5) / P(5) = (37/38) / (1/38) = 37:1. Conversely, the odds in favor of landing on 5 are calculated as:

  • Odds in favor of an event = P(event) / P(not event) = P(5) / P(not 5) = (1/38) / (37/38) = 1:37.

Probability of Dice Rolls

• Rolling a Die: When rolling a fair six-sided die, the total possible outcomes are {1, 2, 3, 4, 5, 6} (6 total outcomes).

  • The probability of rolling a 2 or 3 can be calculated as:

    • P(2 or 3) = 2 favorable outcomes / 6 total outcomes = 1/3.

  • Meanwhile, the probability of rolling a 2, 3, or 4 is:

    • P(2 or 3 or 4) = 3 favorable outcomes / 6 total outcomes = 1/2.

Probability of Coin Tosses

• Tossing Coins: The probabilities associated with tossing coins can be derived from the basic outcomes. For example, the probability of getting heads (H) twice in two tosses is:

  • P(H,H) = P(H) * P(H) = (1/2) * (1/2) = 1/4.

  • Similarly, the probability of getting HHH in three tosses is:

  • P(H,H,H) = (1/2)^3 = 1/8.

Card Probabilities

• Drawing from a Deck: In a standard deck of 52 cards, there are 13 spades and 4 kings. However, the King of Spades is accounted for in both categories.

  • Therefore, the probability of drawing either a spade or a king from the deck is:

    • P(spade or king) = (13 + 3) / 52 = 16/52 = 4/13.

  • The specific probability of initially drawing the King of Diamonds is:

    • P(King of Diamonds) = 1/52.

  • If the King of Diamonds is drawn first, the probability of drawing another king follows as:

    • P(King 2nd | King 1st not replaced) = 3/51.

  • For the joint probability of picking the King of Diamonds followed by another king, we can use:

    • P(King 1st and King 2nd) = (1/52)*(3/51) = 3/2652.

Probability of Defects

• Assessing Product Quality: Evaluating the probability of receiving at least one defective product when purchasing 12 items with a defect rate of 15% can be done using the complement rule.

  • Define:

    • P(no defective) = (1 - 0.15)^12 = (0.85)^12. Hence, the probability of at least one defective product is:

    • P(at least one defective) = 1 - (0.85)^12.

Dish Orders at a Restaurant

• Understanding Ordering Patterns: Analyzing dish orders based on gender, where 60% of the patrons are men and 40% are women, reveals:

  • 80% of men order Dish A; thus, the total men ordering A = 0.6 * 0.8 = 0.48.

  • On the other hand, 70% of women opt for Dish B, leading to women for B = 0.4 * 0.7 = 0.28.

  • The total orders for Dish A include contributions from both genders:

    • Total for A = 0.48 (men) + 0.12 (women who order Dish A) = 0.60.

  • For Dish B, it totals:

    • Total for B = 0.28 (from women) + 0.12 (men) = 0.40.

Probability of Scooter Quality from Plants

• Evaluating Manufacturing Standards: Considering the quality of scooters produced by two plants, where Plant I manufactures 80% of the scooters and maintains a standard quality rate of 85%, while Plant II makes 20% with a quality rate of 65%, we apply Bayes' theorem to derive probabilities:

  • To find P(Plant I | Standard Quality), calculate:

    • P(Standard) = P(Plant I) * P(Standard | Plant I) + P(Plant II) * P(Standard | Plant II).

Cumulative Distribution Function (CDF)

• CDF Definition and Use: The cumulative distribution function, denoted as FX(x), measures the probability that a random variable X is less than or equal to x. The properties include:

  • 0 ≤ FX(x) ≤ 1, FX(∞) = 1, FX(−∞) = 0, and it is a non-decreasing function: FX(x1) ≤ FX(x2) if x1 ≤ x2.

  • Probabilities can be derived using:

    • P(a ≤ X ≤ b) = FX(b) − FX(a).

Probability Density Function (PDF)

• Understanding PDFs: The probability density function, fX(x), serves as the derivative of the cumulative distribution function (CDF). Its properties ensure:

  • fX(x) ≥ 0, and the integral over the complete space equals 1: ∫ fX(x) dx = 1.

  • For example, determining a constant k for a given PDF such as fX(x) = kx for 0 < x < 1 requires setting appropriate conditions based on the properties of PDFs.

Mean and Variance of Random Variables

• Calculating Mean and Variance: For a random variable X with a given PDF, the expected value or mean is computed as:

  • E(X) = ∫ x * fX(x) dx.

  • To find the variance of X, use:

  • Var(X) = E(X^2) - (E(X))^2, where E(X^2) = ∫ x^2 * fX(x) dx.