# Confusing questions about Binomial Distributions

• Jun 4th 2009, 10:48 PM
krzyrice

I seriously don't know where to start on number 7. And can you confirm that 6a) is correct. Thank you so much!
• Jun 4th 2009, 11:10 PM
mr fantastic
Quote:

Originally Posted by krzyrice

I seriously don't know where to start on number 7. And can you confirm that 6a) is correct. Thank you so much!

Please type the questions out. It's annoying and inconvenient to have to click on a link.
• Jun 4th 2009, 11:28 PM
krzyrice
2. Show that for a binomial distribution B with a fixed number of trials n and a fixed probability p = 0.5, B(k) = B(n - k).

6. Suppose you flip a quarter 16 times and count the times it lands heads up.
a) What is the expected number of heads? 8?
b) What is the probability that the number of heads will be no more than one standard deviation from from the expected number?

7.Suppose the following experiment is conducted: A die with s sides marked 1 through s is tossed n times and the number of times a 1 is tossed is recorded. After many repetitions of the experiment, it is found that the number of 1s tossed has a mean of 33 and a standard deviation of 5.5.
a) If the die is assumed to be fair, what is the most likely number of sides it has?
b) If the die is assumed to be fair, how many times n was tossed in each experiment?
• Jun 5th 2009, 02:47 AM
mr fantastic
Quote:

Originally Posted by krzyrice
2. Show that for a binomial distribution B with a fixed number of trials n and a fixed probability p = 0.5, B(k) = B(n - k).

6. Suppose you flip a quarter 16 times and count the times it lands heads up.
a) What is the expected number of heads? 8?
b) What is the probability that the number of heads will be no more than one standard deviation from from the expected number?

[snip]

Q2. You should know that $B(x) = {n \choose x} p^x (1 - p)^{n - x} = {n \choose x} (0.5)^x (0.5)^{n - x}$. Substitute $x = k$ and $x = n - k$. What do you get?

Q6. X ~ Binomial(n = 16, p = 0.5).

(b) $\sigma = \sqrt{(16)(0.5)(0.5)} = 2$. So calculate $\Pr(6 \leq X \leq 10)$.

Quote:

Originally Posted by krzyrice
[snip]
7.Suppose the following experiment is conducted: A die with s sides marked 1 through s is tossed n times and the number of times a 1 is tossed is recorded. After many repetitions of the experiment, it is found that the number of 1s tossed has a mean of 33 and a standard deviation of 5.5.
a) If the die is assumed to be fair, what is the most likely number of sides it has?
b) If the die is assumed to be fair, how many times n was tossed in each experiment?

Let X be the random variable number of times a 1 is tossed.

X ~ Binomial(n, p = 1/s).

$\mu = np \Rightarrow 33 = \frac{n}{s}$ .... (1)

$\sigma^2 = n p (1 - p) \Rightarrow 5.5^2 = \frac{n (s - 1)}{s^2}$ .... (2)

Substitute (1) into (2) and solve for $s$. Hence solve for $n$.