A Fair Coin Is Tossed Six Times. Find The Following Probabilities:a) Getting All Headsb) The Tosses Are The Same
Introduction
In probability theory, a fair coin is a coin that has an equal chance of landing on either heads or tails. When a fair coin is tossed, the probability of getting heads or tails is 1/2 or 0.5. In this article, we will explore the probabilities of two events when a fair coin is tossed six times: getting all heads and the tosses being the same.
Getting All Heads
To calculate the probability of getting all heads when a fair coin is tossed six times, we need to understand the concept of independent events. When a coin is tossed, the outcome of the next toss is independent of the previous toss. This means that the probability of getting heads or tails on the next toss is not affected by the previous toss.
Since the coin is fair, the probability of getting heads on each toss is 1/2 or 0.5. To calculate the probability of getting all heads, we need to multiply the probability of getting heads on each toss.
Calculating the Probability of Getting All Heads
The probability of getting heads on the first toss is 1/2 or 0.5. The probability of getting heads on the second toss is also 1/2 or 0.5, and so on. Since the tosses are independent, we can multiply the probabilities of getting heads on each toss to get the probability of getting all heads.
P(all heads) = P(heads on 1st toss) × P(heads on 2nd toss) × P(heads on 3rd toss) × P(heads on 4th toss) × P(heads on 5th toss) × P(heads on 6th toss) = 0.5 × 0.5 × 0.5 × 0.5 × 0.5 × 0.5 = 0.03125
Therefore, the probability of getting all heads when a fair coin is tossed six times is 0.03125 or 3.125%.
The Tosses Are the Same
To calculate the probability of the tosses being the same, we need to consider two cases: all heads and all tails.
Calculating the Probability of All Heads and All Tails
We have already calculated the probability of getting all heads, which is 0.03125 or 3.125%. To calculate the probability of getting all tails, we can use the same formula:
P(all tails) = P(tails on 1st toss) × P(tails on 2nd toss) × P(tails on 3rd toss) × P(tails on 4th toss) × P(tails on 5th toss) × P(tails on 6th toss) = 0.5 × 0.5 × 0.5 × 0.5 × 0.5 × 0.5 = 0.03125
Since the probability of getting all heads and all tails are the same, we can add them together to get the probability of the tosses being the same:
P(tosses are the same) = P(all heads) + P(all tails) = 0.03125 + 0.03125 = 0.0625
Therefore, the probability of the tosses being the same when a fair coin is tossed six times is 0.0625 or 6.25%.
Conclusion
In this article, we calculated the probabilities of two events when a fair coin is tossed six times: getting all heads and the tosses being the same. We used the concept of independent events and the formula for multiplying probabilities to calculate the probability of getting all heads. We also considered two cases: all heads and all tails, to calculate the probability of the tosses being the same. The results show that the probability of getting all heads is 3.125% and the probability of the tosses being the same is 6.25%.
References
- [1] Probability Theory, by E.T. Jaynes
- [2] Statistics, by W.H. Kruskal and J.M. Tanur
Glossary
- Independent events: Events that do not affect each other.
- Probability: A measure of the likelihood of an event occurring.
- Fair coin: A coin that has an equal chance of landing on either heads or tails.
A Fair Coin Tossed Six Times: Q&A =====================================
Introduction
In our previous article, we explored the probabilities of two events when a fair coin is tossed six times: getting all heads and the tosses being the same. In this article, we will answer some frequently asked questions related to the topic.
Q: What is the probability of getting at least one head when a fair coin is tossed six times?
A: To calculate the probability of getting at least one head, we need to subtract the probability of getting all tails from 1. The probability of getting all tails is 0.03125, as calculated earlier. Therefore, the probability of getting at least one head is:
P(at least one head) = 1 - P(all tails) = 1 - 0.03125 = 0.96875
Q: What is the probability of getting exactly two heads when a fair coin is tossed six times?
A: To calculate the probability of getting exactly two heads, we need to use the binomial distribution formula. The binomial distribution formula is:
P(x) = (n choose x) × p^x × (1-p)^(n-x)
where n is the number of trials, x is the number of successes, p is the probability of success, and (n choose x) is the number of combinations of n items taken x at a time.
In this case, n = 6, x = 2, and p = 0.5. Therefore, the probability of getting exactly two heads is:
P(2 heads) = (6 choose 2) × 0.5^2 × 0.5^4 = 15 × 0.25 × 0.0625 = 0.234375
Q: What is the probability of getting at most two heads when a fair coin is tossed six times?
A: To calculate the probability of getting at most two heads, we need to add the probabilities of getting 0, 1, and 2 heads. We have already calculated the probability of getting 0 heads (all tails) and 2 heads. The probability of getting 1 head is:
P(1 head) = (6 choose 1) × 0.5^1 × 0.5^5 = 6 × 0.5 × 0.03125 = 0.09375
Therefore, the probability of getting at most two heads is:
P(at most 2 heads) = P(0 heads) + P(1 head) + P(2 heads) = 0.03125 + 0.09375 + 0.234375 = 0.359375
Q: What is the probability of getting more than two heads when a fair coin is tossed six times?
A: To calculate the probability of getting more than two heads, we need to subtract the probability of getting at most two heads from 1. Therefore, the probability of getting more than two heads is:
P(more than 2 heads) = 1 - P(at most 2 heads) = 1 - 0.359375 = 0.640625
Q: What is the probability of getting an even number of heads when a fair coin is tossed six times?
A: To calculate the probability of getting an even number of heads, we need to add the probabilities of getting 0, 2, 4, and 6 heads. We have already calculated the probabilities of getting 0 and 2 heads. The probability of getting 4 heads is:
P(4 heads) = (6 choose 4) × 0.5^4 × 0.5^2 = 15 × 0.0625 × 0.25 = 0.046875
The probability of getting 6 heads is:
P(6 heads) = (6 choose 6) × 0.5^6 = 1 × 0.015625 = 0.015625
Therefore, the probability of getting an even number of heads is:
P(even number of heads) = P(0 heads) + P(2 heads) + P(4 heads) + P(6 heads) = 0.03125 + 0.234375 + 0.046875 + 0.015625 = 0.328125
Conclusion
In this article, we answered some frequently asked questions related to the topic of a fair coin tossed six times. We used the binomial distribution formula to calculate the probabilities of getting exactly two heads, at most two heads, and more than two heads. We also calculated the probability of getting an even number of heads. The results show that the probability of getting at least one head is 96.875%, the probability of getting exactly two heads is 23.4375%, and the probability of getting an even number of heads is 32.8125%.
References
- [1] Probability Theory, by E.T. Jaynes
- [2] Statistics, by W.H. Kruskal and J.M. Tanur
Glossary
- Binomial distribution: A discrete probability distribution that models the number of successes in a fixed number of independent trials.
- Independent trials: Trials that do not affect each other.
- Probability of success: The probability of getting a success in a single trial.
- Probability of failure: The probability of getting a failure in a single trial.