Multiplication Rule and Probability

Objectives

  • Compute probabilities using the Multiplication Rule for Independent Events.

Independence of Events

  • Two events are independent if the occurrence of one does not affect the other.

  • Events are dependent if the outcome of one impacts the other.

  • Examples:

    • Drawing a card and rolling a die (independent).

    • Drawing cards without replacement (dependent).

Multiplication Rule for Independent Events

  • For independent events A and B, the probability is given by:
    P(B|A) = P(B)

  • Therefore, the Multiplication Rule states:
    P(A\vert B)=P(A)\cdot P\left(B\right)

Example of Independent Events

  • Probability of all selected persons under 18 from different cities:

    • New York: 23.5%, Chicago: 25.8%, Los Angeles: 26.0%

    • Calculation: P(All<18)=0.235\cdot0.258\cdot0.260=0.0158

    • This is considered an unusual event (cutoff point of 0.05).

Example: Survival Probability

  • Probability a randomly selected 60-year-old female survives the year is 99.186%.

  • For four females, the probability is:
    P(all 4 survive) = (0.99186)^4 = 0.9678

Sampling Methods

  • Sampling with Replacement: Independent events since each draw is from the whole population.

  • Sampling without Replacement: Dependent events since the first draw affects the second.

  • Rule of Thumb: Treat as independent if sample is less than 5% of the population.

Knowledge Check: NCLB Act

  • Each student has a 0.99 probability of passing.

  • Questions: Probability everyone passes vs. probability 3 pass in a school of 500 students.

Probability Calculation

For a telecommunications company with 7 satellites (3 weak signals), the probability of picking 2 weak signals without replacement needs to be calculated.

  • Probability of all selected persons under 18 from different cities:

    • New York: 23.5%, Chicago: 25.8%, Los Angeles: 26.0%

    • Calculation: P(All<18)=0.235\cdot0.258\cdot0.260=0.0158

    • This is considered an unusual event (cutoff point of 0.05).

New Example: Coin Flips and Dice Rolls

Let's calculate the probability of flipping a coin and getting a 'Heads' AND rolling a standard six-sided die and getting a '5'.

Step-by-step solution:

  1. Identify the events:

    • Event A: Flipping a coin and getting 'Heads'.

    • Event B: Rolling a standard six-sided die and getting a '5'.

  2. Determine independence:

    • Flipping a coin and rolling a die are completely separate actions. The outcome of the coin flip does not in any way influence the outcome of the die roll, and vice-versa. Therefore, these events are independent.

  3. Calculate individual probabilities:

    • Probability of Event A (P(A)) = Probability of getting 'Heads' = 1/2 = 0.5

    • Probability of Event B (P(B)) = Probability of getting a '5' on a six-sided die = 1/6 \approx 0.1667

  4. Apply the Multiplication Rule for Independent Events:

    • Since the events are independent, we use the formula: P(A \text{ and } B) = P(A) \cdot P(B)

    • P(\text{Heads and 5}) = (1/2) \cdot (1/6) = 1/12

    • In decimal form: 0.5 \cdot 0.1667 \approx 0.08335

Reasoning for this method:

We use the Multiplication Rule for Independent Events because the act of flipping a coin has absolutely no bearing on the act of rolling a die. If these events were dependent (e.g., drawing two cards without replacement from a deck, where the first draw changes the probabilities for the second draw), we would need to use a conditional probability formula, such as P(A \text{ and } B) = P(A) \cdot P(B|A). However, in this case, the probability of rolling a 5 is the same regardless of whether the coin landed on heads or tails, demonstrating their independence and simplifying the calculation to a direct product of individual probabilities.

Knowledge check:

A telecommunications company has seven satellites, three of which are sending a weak signal. If two are picked at random without replacement, find the probability that both are sending a weak signal.

Express your answer as a fraction in lowest terms.