Independent Events and Probability

• Independent events occur when the chance of one event happening does not affect the chance of another event happening

Probability of an Event

• P = e/n
• P represents the probability of an event happening.
• e represents the number of ways an event can happen.
• n represents the total number of possible outcomes.
• Example: In a basket with 4 oranges, 3 pears, and 3 apples, the probability of picking an apple is 3/10

Multiplication Rule for Independent Events

• The probability of multiple independent events occurring is the product of their individual probabilities
• Indicator word "and" indicates multiplication
• P = PA * PB * PC * ...

Checking for Independence

• If P(A and B) is known, events A and B are independent if PA * PB = P(A and B)

Example of Multiple Independent Events

• In a basket with 4 oranges, 3 pears, and 3 apples, and a jar with 8 chocolate, 8 coffee, and 4 caramel candies, the probability of picking an apple and a chocolate candy is (3/10) * (8/20) = 0.12

Probability of at Least One Event

• Finding the probability of at least one event happening out of multiple independent events can be done with two methods:
  • Complementary Probability
  • Probability of Repeated Trials

Method A: Complementary Probability

• Find the probability of something not happening, rather than the probability that will happen

• Step 1: Find the complementary probability of a successful independent event.

• Step 2: Apply the multiplication rule on the complementary probability of successful independent events.

• Step 3: Solve for the probability of at least one event by finding the complementary probability of the multiple events.

• P(at least one success) = 1−P(all failures)

Method B: Probability of Repeated Trials

• Find the summation of the probability of repeated trials.
• P(x successes in n trials) = (n! / (x! * (n-x)!)) * s^x * f^(n-x)
  • n is the number of trials
  • s is the success probability
  • f is the fail or complementary probability

Example 1: Coin Toss

• A coin is tossed three times; determine the probability of at least one tail.
• Method A:
  • Probability of not getting a tail (complementary probability) = 1/2
  • Probability of not getting a tail in three tosses = (1/2) * (1/2) * (1/2) = 1/8
  • Probability of at least one tail = 1 - (1/8) = 7/8
• Method B:
  • P(1 tail) + P(2 tails) + P(3 tails) = 3/8 + 3/8 + 1/8 = 7/8

Example 2: Drawing Cards

• Two cards are drawn from a deck, with the first card returned before the second draw; what is the probability of drawing at least one queen?
  • Complementary probability of not drawing a queen = 48/52
  • Probability of not drawing a queen in two picks = (48/52) * (48/52)
  • Probability of at least one queen = 1 - ((48/52) * (48/52)) ≈ 0.148

Example 3: Passing an Exam

• The probability of passing a math exam is 4/5; if three students take the exam, what is the probability of at least one student passing?
  • Uses Method B
  • P(1 success) + P(2 successes) + P(3 successes)
  • (3!/(1!2!)) * (4/5)^1 * (1/5)^2 + (3!/(2!1!)) * (4/5)^2 * (1/5)^1 + (3!/(3!0!)) * (4/5)^3 * (1/5)^0 = 0.992

Example 4: Horse Race

• Fe, Pio, and Nana bet on a horse race with a 1/10 chance of winning; what is the probability that at least one of them will win?
  • Probability of not winning for each person = 9/10 = 0.9
  • Probability of all 3 not winning = 0.9 * 0.9 * 0.9 = 0.729
  • Probability of at least one winning = 1 - 0.729 = 0.271

Understand independent events and calculate probabilities. Learn the multiplication rule for independent events. Includes an example calculation with fruits and candies.