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19. Five fair coins are each flipped once. the probability that at least two of the coins will show heads? What is

19. Five fair coins are each flipped once. the probability that at least two of the coins will show heads? What is

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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please explain in detail. the question is placed in as picture.

also if there is any kind of shortcut formula for such problems 

**Problem 19:** Five fair coins are each flipped once. What is the probability that at least two of the coins will show heads? This problem involves calculating the probability of a specific outcome when flipping five fair coins. The coins are unbiased, meaning each has a 50% chance of landing heads (H) and a 50% chance of landing tails (T). To find the probability of at least two coins showing heads, consider the following: 1. **Total Outcomes:** Each coin flip has two possible outcomes: heads or tails. Therefore, the total number of possible outcomes when flipping five coins is \(2^5 = 32\). 2. **Favorable Outcomes:** We need to count the number of outcomes where at least two coins are heads. This can be found by subtracting the scenarios where fewer than two coins show heads: - **No heads (0 H):** Only one outcome (TTTTT). - **One head (1 H):** There are 5 different outcomes (e.g., HTTTT, THTTT, etc.). 3. **Calculation:** - Outcomes with 0 or 1 head: \(1 + 5 = 6\). - Outcomes with at least 2 heads: \(32 - 6 = 26\). 4. **Probability:** - Probability of at least two heads = \(\frac{26}{32} = \frac{13}{16}\). The probability that at least two of the five coins will show heads is \(\frac{13}{16}\).
Transcribed Image Text:**Problem 19:** Five fair coins are each flipped once. What is the probability that at least two of the coins will show heads? This problem involves calculating the probability of a specific outcome when flipping five fair coins. The coins are unbiased, meaning each has a 50% chance of landing heads (H) and a 50% chance of landing tails (T). To find the probability of at least two coins showing heads, consider the following: 1. **Total Outcomes:** Each coin flip has two possible outcomes: heads or tails. Therefore, the total number of possible outcomes when flipping five coins is \(2^5 = 32\). 2. **Favorable Outcomes:** We need to count the number of outcomes where at least two coins are heads. This can be found by subtracting the scenarios where fewer than two coins show heads: - **No heads (0 H):** Only one outcome (TTTTT). - **One head (1 H):** There are 5 different outcomes (e.g., HTTTT, THTTT, etc.). 3. **Calculation:** - Outcomes with 0 or 1 head: \(1 + 5 = 6\). - Outcomes with at least 2 heads: \(32 - 6 = 26\). 4. **Probability:** - Probability of at least two heads = \(\frac{26}{32} = \frac{13}{16}\). The probability that at least two of the five coins will show heads is \(\frac{13}{16}\).
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