Question

Label each situation/experiment as

binomial, (B)
geometric (G)
neither. (N)

1) A card is selected from a standard deck and replaced. This experiment is repeated
8 times. Find the probability of selecting exactly three clubs.

2) The probability of getting a defective computer is 0.2. Find the probability that the first defect will be is in the seventh computer tested

3) Selecting 5 cards from a standard deck, without replacement, and recording the number of hearts.

4) A coin is tossed five times and the number of tails is recorded.

5) A basketball player shoots free throws until he makes one.

6) It is estimated that 87% of computer users use Explorer as their default web browser. We choose 50 computer users and ask their default browser.

Answer 1

1) Probability of Selecting Exactly Three Clubs: Probability: ≈ 0.2504. Label: (B). 2) Probability of First Defect on the Seventh Computer: Probability: ≈ 0.0262. Label: (G). 3) Selecting 5 Cards, Recording the Number of Hearts: Probability: ≈ 0.0588. Label: (N). 4) Number of Tails in 5 Coin Tosses: Probabilities for k = 0, 1, 2, 3, 4, 5: ≈ 0.03125, 0.15625, 0.3125, 0.3125, 0.15625, 0.03125. Label: (B). 5) Basketball Player Shooting Until Making One: Probabilities for k = 1, 2, 3,...: ≈ 0.5, 0.25, 0.125,... Label: (G). 6) Default Web Browser of Computer Users: Probabilities for k = 0, 1, 2, ..., 50: Vary based on the specific value of k. Label: (B)

1) Probability of Selecting Exactly Three Clubs:

- Situation: Binomial (B)

- Probability Formula:
`\({n \choose k} \cdot p^k \cdot (1-p)^(n-k)\)`

- Calculation:
`\({8 \choose 3} \cdot \left((1)/(4)\right)^3 \cdot \left((3)/(4)\right)^5\)` ≈ 0.2504

2) Probability of First Defect on the Seventh Computer:

- Situation: Geometric (G)

- Probability Formula:
`\(P(X=k) = (1-p)^(k-1) \cdot p\)`

- Calculation:
`\((0.8)^6 \cdot 0.2\)` ≈ 0.0262144

3) Selecting 5 Cards, Recording the Number of Hearts:

- Situation: Neither (N) - It involves multiple trials, but the probability changes after each draw.

- Probability Calculation:
`\(\frac{{13 \choose 5}}{{52 \choose 5}}\)` ≈ 0.0588

4) Number of Tails in 5 Coin Tosses:

- Situation: Binomial (B)

- Probability Formula:
`\({n \choose k} \cdot p^k \cdot (1-p)^(n-k)\)`

- Calculation:
`\({5 \choose k} \cdot \left((1)/(2)\right)^k \cdot \left((1)/(2)\right)^(5-k)\) for \(k = 0, 1, 2, 3, 4, 5\)`

≈ 0.03125, 0.15625, 0.3125, 0.3125, 0.15625, 0.03125

5) Basketball Player Shooting Until Making One:

- Situation: Geometric (G)

- Probability Formula:
`\(P(X=k) = (1-p)^(k-1) \cdot p\)`

- Calculation:
`\(0.5^k \cdot 0.5\) for \(k = 1, 2, 3, \ldots\)`

≈ 0.5, 0.25, 0.125,…

6) Default Web Browser of Computer Users:

- Situation: Binomial (B)

- Probability Formula:
`\({n \choose k} \cdot p^k \cdot (1-p)^(n-k)\)`

- Calculation:
`\({50 \choose k} \cdot (0.87)^k \cdot (0.13)^(50-k)\) for \(k = 0, 1, 2, \ldots, 50\)`

Vary based on the specific value of k.