6.3 Cumulative Probabilities- targeted

Key Facts of Cumulative Probabilities

• Definition of Cumulative Probability: For a random variable X following a binomial distribution with n trials and probability P of success, the cumulative probability is the probability that X is less than or equal to a specific value r.

• General Form: The probability that X is less than or equal to r is denoted as P(X \leq r).

• Calculator Usage: Cumulative probabilities are computed using the binomial cumulative distribution (CD) function on a calculator.

Understanding Inequalities

• Probability of X Less Than r: If we want to find the probability that X is strictly less than r, we consider P(X < r). This is equivalent to P(X \leq r-1). In general:
  P(X < r) = P(X \leq r-1)

  Example:

  P(X < 5) = P(X \leq 4)

• Probability of X Greater Than r: If we want to find the probability that X is greater than r, we consider P(X > r). This is equivalent to 1 - P(X \leq r). The value r is included in P(X \leq r) because it is not included in P(X > r). In general:
  P(X > r) = 1 - P(X \leq r)

  Example:

  P(X > 5) = 1 - P(X \leq 5)

• Probability of X Greater Than or Equal to r: If we want to find the probability that X is greater than or equal to r, we consider P(X \geq r). This is equivalent to 1 - P(X \leq r-1). The value r is included in P(X \geq r). In general:
  P(X \geq r) = 1 - P(X \leq r-1)

  Example:

  P(X \geq 5) = 1 - P(X \leq 4)

Exam-Style Question 1

• Problem: In a particular type of plant, 25% have blue flowers. A garden center sells these plants in trays of 15 plants of mixed colors. A tray is selected at random. Find the probability that the number of plants with blue flowers in this tray is:

  • Exactly 4.
  • At most 3.
  • Between 3 and 6 inclusive.

• Define Random Variable: Let X be the number of plants with blue flowers.

• Distribution: X follows a binomial distribution with n = 15 trials and p = 0.25 probability of success (blue flower).

  X \sim B(15, 0.25)

Part A: Probability of Exactly 4 Blue Flowers

• Task: Find P(X = 4).

• Method: Use the binomial probability distribution (PD) function on the calculator.

• Calculator Input: n = 15, p = 0.25, x = 4.

• Result: P(X = 4) = 0.2252 (to four decimal places).

Part B: Probability of At Most 3 Blue Flowers

• Task: Find P(X \leq 3). "At most 3" means 3 or less.

• Method: Use the binomial cumulative distribution (CD) function on the calculator.

• Calculator Input: n = 15, p = 0.25, x = 3.

• Result: P(X \leq 3) = 0.4613 (to four decimal places).

Part C: Probability Between 3 and 6 Inclusive

• Task: Find P(3 \leq X \leq 6).

• Method: Use the binomial cumulative distribution (CD) function on the calculator.

• Express P(3 \leq X \leq 6) as the probability that X is less than or equal to 6 minus the probability that X is less than or equal to 2: P(3 \leq X \leq 6) = P(X \leq 6) - P(X \leq 2)

• Calculator Input for P(X \leq 6): n = 15, p = 0.25, x = 6.

  • P(X \leq 6) = 0.9434 (to four decimal places).

• Calculator Input for P(X \leq 2): n = 15, p = 0.25, x = 2.

  • P(X \leq 2) = 0.2361 (to four decimal places).

• Final Calculation: P(3 \leq X \leq 6) = 0.9434 - 0.2361 = 0.7073.

Binomial Cumulative Distribution Function Table

• Table Usage: Valid only for probabilities of the form P(X \leq x).

• Example: If X \sim B(5, 0.10), to find P(X \leq 3), look up the value corresponding to x = 3 in the table for n = 5 and p = 0.10.

Exam-Style Question 2

• Problem: The random variable X takes on a binomial distribution with 40 trials and fixed probability 0.10. Find

  • The largest value of K such that P(X < K) < 0.02.
  • The smallest value of R such that P(X > R) < 0.01.
  • P(K \leq X \leq R).

• Distribution: X \sim B(40, 0.10).

Part A: Finding the Largest Value of K

• Task: Find the largest integer K such that P(X < K) < 0.02.

• Rewrite the Inequality: Since the table is for P(X \leq x), rewrite P(X < K) as P(X \leq K-1). So, P(X \leq K-1) < 0.02.

• Table Lookup: Look in the binomial cumulative distribution function table with n = 40 and p = 0.10 to find the largest value of K-1 such that the cumulative probability is less than 0.02. The table gives K-1 = 0.

• Solve for K: Since K-1 = 0, K = 1.

Part B: Finding the Smallest Value of R

• Task: Find the smallest integer R such that P(X > R) < 0.01.

• Rewrite the Inequality: Rewrite P(X > R) as 1 - P(X \leq R) < 0.01.

  • Rearrange this inequality:
    1 - P(X \leq R) < 0.01 - P(X \leq R) < -0.99 P(X \leq R) > 0.99

• Table Lookup: Look in the binomial cumulative distribution function table with n = 40 and p = 0.10 to find the smallest value of R such that the cumulative probability is greater than 0.99. From the table, R = 9.

Part C: Calculating P(K \leq X \leq R)

• Task: Calculate P(1 \leq X \leq 9), given K = 1 and R = 9.

• Method: Express P(1 \leq X \leq 9) as the probability that X is less than or equal to 9 minus the probability that X is less than or equal to 0: P(1 \leq X \leq 9) = P(X \leq 9) - P(X \leq 0)

• Table Lookup: From the binomial cumulative distribution function table with n = 40 and p = 0.10, find P(X \leq 9) and P(X \leq 0).

  • P(X \leq 9) = 0.9949
  • P(X \leq 0) = 0.0148

• Final Calculation: P(1 \leq X \leq 9) = 0.9949 - 0.0148 = 0.9801.