Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine the probability that you will get "heads" no more than once out of 3 flips, we need to consider the scenarios in which you get 0 or 1 head. We can break this down into two parts: the probability of getting 0 heads and the probability of getting 1 head.
We will use the binomial distribution formula to find these probabilities. For a binomial distribution with [tex]\( n \)[/tex] trials and probability [tex]\( p \)[/tex] of success on each trial, the probability of getting exactly [tex]\( k \)[/tex] successes is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, representing the number of ways to choose [tex]\( k \)[/tex] successes out of [tex]\( n \)[/tex] trials, and is calculated as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
Given:
- [tex]\( n = 3 \)[/tex] (the number of flips)
- [tex]\( p = 0.5 \)[/tex] (the probability of getting heads in a single flip)
1. Calculate the probability of getting 0 heads ([tex]\( k = 0 \)[/tex]):
[tex]\[ P_0 = \binom{3}{0} (0.5)^0 (1 - 0.5)^{3-0} \][/tex]
[tex]\[ P_0 = \frac{3!}{0!(3-0)!} (0.5)^0 (0.5)^3 \][/tex]
[tex]\[ P_0 = 1 \cdot 1 \cdot (0.5)^3 \][/tex]
[tex]\[ P_0 = 1 \cdot 0.125 \][/tex]
[tex]\[ P_0 = 0.125 \][/tex]
2. Calculate the probability of getting 1 head ([tex]\( k = 1 \)[/tex]):
[tex]\[ P_1 = \binom{3}{1} (0.5)^1 (1 - 0.5)^{3-1} \][/tex]
[tex]\[ P_1 = \frac{3!}{1!(3-1)!} (0.5)^1 (0.5)^2 \][/tex]
[tex]\[ P_1 = 3 \cdot 0.5 \cdot (0.5)^2 \][/tex]
[tex]\[ P_1 = 3 \cdot 0.5 \cdot 0.25 \][/tex]
[tex]\[ P_1 = 3 \cdot 0.125 \][/tex]
[tex]\[ P_1 = 0.375 \][/tex]
Adding the probabilities together:
[tex]\[ P(\text{0 or 1 heads}) = P_0 + P_1 \][/tex]
[tex]\[ P(\text{0 or 1 heads}) = 0.125 + 0.375 \][/tex]
[tex]\[ P(\text{0 or 1 heads}) = 0.5 \][/tex]
Therefore, the probability that you will get "heads" no more than once out of 3 flips is [tex]\( \boxed{0.5} \)[/tex].
We will use the binomial distribution formula to find these probabilities. For a binomial distribution with [tex]\( n \)[/tex] trials and probability [tex]\( p \)[/tex] of success on each trial, the probability of getting exactly [tex]\( k \)[/tex] successes is given by:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
where [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, representing the number of ways to choose [tex]\( k \)[/tex] successes out of [tex]\( n \)[/tex] trials, and is calculated as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \][/tex]
Given:
- [tex]\( n = 3 \)[/tex] (the number of flips)
- [tex]\( p = 0.5 \)[/tex] (the probability of getting heads in a single flip)
1. Calculate the probability of getting 0 heads ([tex]\( k = 0 \)[/tex]):
[tex]\[ P_0 = \binom{3}{0} (0.5)^0 (1 - 0.5)^{3-0} \][/tex]
[tex]\[ P_0 = \frac{3!}{0!(3-0)!} (0.5)^0 (0.5)^3 \][/tex]
[tex]\[ P_0 = 1 \cdot 1 \cdot (0.5)^3 \][/tex]
[tex]\[ P_0 = 1 \cdot 0.125 \][/tex]
[tex]\[ P_0 = 0.125 \][/tex]
2. Calculate the probability of getting 1 head ([tex]\( k = 1 \)[/tex]):
[tex]\[ P_1 = \binom{3}{1} (0.5)^1 (1 - 0.5)^{3-1} \][/tex]
[tex]\[ P_1 = \frac{3!}{1!(3-1)!} (0.5)^1 (0.5)^2 \][/tex]
[tex]\[ P_1 = 3 \cdot 0.5 \cdot (0.5)^2 \][/tex]
[tex]\[ P_1 = 3 \cdot 0.5 \cdot 0.25 \][/tex]
[tex]\[ P_1 = 3 \cdot 0.125 \][/tex]
[tex]\[ P_1 = 0.375 \][/tex]
Adding the probabilities together:
[tex]\[ P(\text{0 or 1 heads}) = P_0 + P_1 \][/tex]
[tex]\[ P(\text{0 or 1 heads}) = 0.125 + 0.375 \][/tex]
[tex]\[ P(\text{0 or 1 heads}) = 0.5 \][/tex]
Therefore, the probability that you will get "heads" no more than once out of 3 flips is [tex]\( \boxed{0.5} \)[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.