# Thread: coin - prob distribution

1. ## coin - prob distribution

I need help with the following question:

A coin is biased to show heads twice as often as it shows tails. You toss this coin 3 times and win 1 dollar each time it shows tails but lose 50 cents each time it shows heads.

a) What is the probability distribution of X: the amount of money you could win or lose in 3 tosses?

b) On average, how much money would you win or lose each time you toss this coin 3 times?

c) Express the maximum amount of money you could win in 3 tosses of this coin as a Z-score?

a) X{-1.50, 0, 1.50, 3.00}, p(X) {0.2963, 0.4444, 0.2222, 0.037}
b) mu = 0
c) Z=2.45

2. Originally Posted by aptiva
I need help with the following question:

A coin is biased to show heads twice as often as it shows tails. You toss this coin 3 times and win 1 dollar each time it shows tails but lose 50 cents each time it shows heads.

a) What is the probability distribution of X: the amount of money you could win or lose in 3 tosses?
Let $\displaystyle x$ represent the number of heads. Then,
$\displaystyle \left\{ \begin{array}{cc}x&P(x)\\0&.0037\\1&.2222\\2&.4444 \\3&.2963$
Thus,
probability of
losing 1.50 is .0037
gaining nothing is .2222
winning 1.50 is .4444
wining 3 is .2963

3. but how are those numbers/probabilities calculated?

4. Originally Posted by aptiva
but how are those numbers/probabilities calculated?
Hello,

you deal with binomial distribution. The probability that the result X happens exactly k-times is with your problem:
$\displaystyle P(X=k)={3\choose k} \cdot \left({2\over3}\right)^k \cdot \left(1-{2\over3}\right)^{(3-k)}$

Plug in the values 0, 1, 2, 3 for k and you'll get the probabilities which ThePerfectHacker had already calculated.

For instance: k = 0:
$\displaystyle P(X=0)={3\choose 0} \cdot \left({2\over3}\right)^0 \cdot \left(1-{2\over3}\right)^{(3-0)}$ = $\displaystyle 1 \cdot 1\cdot \left({1\over3}\right)^{3}={1 \over 27} \approx .037$

Greetings

EB