What is the maximum likelihood of getting k heads when you tossed coin n times? Explain the math behind it.
Question Analysis
The question is asking about the probability of obtaining exactly ( k ) heads when a fair coin is tossed ( n ) times. This is a question related to probability and statistics, specifically involving the binomial distribution. The "maximum likelihood" in this context refers to the most probable outcome or the most likely number of heads you would expect to get. Understanding the concept of a binomial distribution is essential, as it models the number of successes in a fixed number of independent Bernoulli trials.
Answer
To determine the maximum likelihood of getting ( k ) heads when tossing a coin ( n ) times, we need to refer to the properties of a binomial distribution. The probability of getting exactly ( k ) heads is given by the binomial probability formula:
[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]
where:
- ( n ) is the total number of tosses,
- ( k ) is the number of heads (successes),
- ( p ) is the probability of getting a head in a single toss,
- ( \binom{n}{k} ) is the binomial coefficient, calculated as ( \frac{n!}{k!(n-k)!} ).
For a fair coin, ( p = 0.5 ). The maximum likelihood occurs at the value of ( k ) that maximizes this probability. For a fair coin, this typically occurs at ( k = \frac{n}{2} ) if ( n ) is even, or at ( k = \left\lfloor \frac{n}{2} \right\rfloor ) or ( k = \left\lceil \frac{n}{2} \right\rceil ) if ( n ) is odd. This is because the binomial distribution is symmetric around ( np ) when ( p = 0.5 ).
Example:
If you toss a coin 10 times (( n = 10 )), the most likely number of heads is ( k = 5 ) because 5 is half of 10, which aligns with the peak of the binomial distribution curve. In this case, the probability of getting exactly 5 heads is calculated using the binomial formula.
By understanding these principles, you can determine the most likely number of heads for any number of coin tosses.