Browse Questions

# Find the probability distribution of the number of sixes in throwing three dice once.

Toolbox:
• If $S$ is a sample space with a probability measure and $X$ is a real valued function defined over the elements of $S$, then $X$ is called a random variable.
• Types of Random variables :
• (1) Discrete Random variable (2) Continuous Random variable
• Discrete Random Variable :If a random variable takes only a finite or a countable number of values, it is called a discrete random variable.
• Continuous Random Variable :A Random Variable X is said to be continuous if it can take all possible values between certain given limits. i.e., X is said to be continuous if its values cannot be put in 1 − 1 correspondence with N, the set of Natural numbers.
• The probability mass function (a discrete probability function) p(x) is a function that satisfies the following properties :
• (1) $P(X=x)=p(x)=p_x$
• (2) $P(x)\geq 0$ for all real $x$
• (3) $\sum p_i=1$
Step 1:
Let $X$ be the discrete RV denoting the number of sixes when three dice are thrown once.
$X$ takes the values $0,1,2,3$
Step 2:
$P(X=0)$=probability that all three dice do not show 6
$\qquad\;\;\;\;\;=\large\frac{5}{6}\times\frac{5}{6}\times\frac{5}{6}=\frac{125}{216}$
$P(X=1)$=probability of one dice showing 6
$\qquad\;\;\;\;\;=3C_1\large\frac{1}{6}\times\frac{5}{6}\times\frac{5}{6}=\frac{75}{216}=\frac{25}{72}$
$P(X=2)$=probability of two dice showing 6
$\qquad\;\;\;\;\;=3C_2\large\frac{1}{6}\times\frac{1}{6}\times\frac{5}{6}=\frac{15}{216}=\frac{5}{72}$
$P(X=3)$=probability of all three dice showing 6
$\qquad\;\;\;\;\;=\large\frac{1}{6}\times\frac{1}{6}\times\frac{1}{6}=\frac{1}{216}$
Step 3:
The probability distribution is given by