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6. A fair coin is tossed three times, and the number of heads is observed. Determine the proba- bility distribution for this experiment, and draw its histogram.

6. A fair coin is tossed three times, and the number of heads is observed. Determine the proba- bility distribution for this experiment, and draw its histogram.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** 6. A fair coin is tossed three times, and the number of heads is observed. Determine the probability distribution for this experiment, and draw its histogram. **Explanation:** To solve this problem, we are considering an experiment where a fair coin is tossed three times. The outcome we are interested in is the number of heads obtained in these three tosses. The possible outcomes for the number of heads are 0, 1, 2, or 3. **Step-by-step Probability Distribution:** 1. **0 Heads:** This occurs when all tosses result in tails (TTT). The probability of this outcome is \( (1/2)^3 = 1/8 \). 2. **1 Head:** This occurs in the following ways: HTT, THT, TTH. Each of these outcomes has a probability of \( 1/8 \), and there are 3 such combinations. Therefore, the probability of getting exactly 1 head is \( 3 \times (1/8) = 3/8 \). 3. **2 Heads:** This occurs in the following ways: HHT, HTH, THH. The probability for each of these outcomes is \( 1/8 \), and there are 3 such combinations. Thus, the probability of getting exactly 2 heads is \( 3 \times (1/8) = 3/8 \). 4. **3 Heads:** This occurs when all tosses result in heads (HHH). The probability of this outcome is \( (1/2)^3 = 1/8 \). **Probability Distribution Table:** | Number of Heads | Probability | |-----------------|-------------| | 0 | 1/8 | | 1 | 3/8 | | 2 | 3/8 | | 3 | 1/8 | **Histogram Explanation:** In the histogram representing this probability distribution: - The x-axis represents the number of heads (0, 1, 2, 3). - The y-axis represents the probability of each outcome. - Each bar's height corresponds to the probability of getting 0, 1, 2, or 3 heads when a fair coin is tossed three times. To visualize this, you can draw bars for each number of heads at heights of 1/8, 3/8, 3/
Transcribed Image Text:**Problem Statement:** 6. A fair coin is tossed three times, and the number of heads is observed. Determine the probability distribution for this experiment, and draw its histogram. **Explanation:** To solve this problem, we are considering an experiment where a fair coin is tossed three times. The outcome we are interested in is the number of heads obtained in these three tosses. The possible outcomes for the number of heads are 0, 1, 2, or 3. **Step-by-step Probability Distribution:** 1. **0 Heads:** This occurs when all tosses result in tails (TTT). The probability of this outcome is \( (1/2)^3 = 1/8 \). 2. **1 Head:** This occurs in the following ways: HTT, THT, TTH. Each of these outcomes has a probability of \( 1/8 \), and there are 3 such combinations. Therefore, the probability of getting exactly 1 head is \( 3 \times (1/8) = 3/8 \). 3. **2 Heads:** This occurs in the following ways: HHT, HTH, THH. The probability for each of these outcomes is \( 1/8 \), and there are 3 such combinations. Thus, the probability of getting exactly 2 heads is \( 3 \times (1/8) = 3/8 \). 4. **3 Heads:** This occurs when all tosses result in heads (HHH). The probability of this outcome is \( (1/2)^3 = 1/8 \). **Probability Distribution Table:** | Number of Heads | Probability | |-----------------|-------------| | 0 | 1/8 | | 1 | 3/8 | | 2 | 3/8 | | 3 | 1/8 | **Histogram Explanation:** In the histogram representing this probability distribution: - The x-axis represents the number of heads (0, 1, 2, 3). - The y-axis represents the probability of each outcome. - Each bar's height corresponds to the probability of getting 0, 1, 2, or 3 heads when a fair coin is tossed three times. To visualize this, you can draw bars for each number of heads at heights of 1/8, 3/8, 3/
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