Answer :
To find the probability of getting at least 7 heads when flipping a fair coin 12 times, we can use the concept of the binomial distribution. Here's how you can calculate it step by step:
1. Understand the Binomial Distribution: The problem involves repeated independent trials (flipping a coin), and we want to find the probability of a certain number of successes (heads) in a given number of trials. This is a classic case for the binomial distribution.
2. Identify the Parameters:
- Number of trials [tex]\( n \)[/tex]: This is the number of times you flip the coin, which is 12.
- Probability of success on each trial [tex]\( p \)[/tex]: Since the coin is fair, the probability of getting heads (success) is 0.5.
- Number of successes [tex]\( k \)[/tex]: We want at least 7 heads, so [tex]\( k \)[/tex] is 7 or more.
3. Calculate the Probability of Getting at Least 7 Heads: To find this, we calculate the probability of getting 7 heads, 8 heads, ..., up to 12 heads, and sum these probabilities. Another approach is to find the cumulative probability of getting fewer than 7 heads and subtract it from 1:
- [tex]\( P(\text{at least 7 heads}) = 1 - P(\text{fewer than 7 heads}) \)[/tex].
4. Using the Binomial Cumulative Distribution Function (CDF): The binomial CDF gives the probability of getting up to a certain number of successes. We calculate [tex]\( P(X < 7) \)[/tex], which is the cumulative probability of getting 0 through 6 heads, and then subtract from 1:
- [tex]\( P(X \geq 7) = 1 - P(X \leq 6) \)[/tex].
5. Result: After performing these calculations, the probability of getting at least 7 heads when flipping a fair coin 12 times is approximately 0.3872.
This means there is about a 38.72% chance of flipping a fair coin 12 times and getting 7 or more heads.
1. Understand the Binomial Distribution: The problem involves repeated independent trials (flipping a coin), and we want to find the probability of a certain number of successes (heads) in a given number of trials. This is a classic case for the binomial distribution.
2. Identify the Parameters:
- Number of trials [tex]\( n \)[/tex]: This is the number of times you flip the coin, which is 12.
- Probability of success on each trial [tex]\( p \)[/tex]: Since the coin is fair, the probability of getting heads (success) is 0.5.
- Number of successes [tex]\( k \)[/tex]: We want at least 7 heads, so [tex]\( k \)[/tex] is 7 or more.
3. Calculate the Probability of Getting at Least 7 Heads: To find this, we calculate the probability of getting 7 heads, 8 heads, ..., up to 12 heads, and sum these probabilities. Another approach is to find the cumulative probability of getting fewer than 7 heads and subtract it from 1:
- [tex]\( P(\text{at least 7 heads}) = 1 - P(\text{fewer than 7 heads}) \)[/tex].
4. Using the Binomial Cumulative Distribution Function (CDF): The binomial CDF gives the probability of getting up to a certain number of successes. We calculate [tex]\( P(X < 7) \)[/tex], which is the cumulative probability of getting 0 through 6 heads, and then subtract from 1:
- [tex]\( P(X \geq 7) = 1 - P(X \leq 6) \)[/tex].
5. Result: After performing these calculations, the probability of getting at least 7 heads when flipping a fair coin 12 times is approximately 0.3872.
This means there is about a 38.72% chance of flipping a fair coin 12 times and getting 7 or more heads.
