Question

Toss $4$times Suppose you toss a fair coin $4$times. Let $X=$the number of heads you get.

(a) Find the probability distribution of$X$.

(b) Make a histogram of the probability distribution. Describe what you see.

(c) Find $P(X\le 3)$ and interpret the result.

Short answer

(a)Probability distribution of $X$is

(b)

(c)$P(X\le 3)=93.75\%$

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that , you toss a fair coin$4$times. Let $X$= the number of heads you get

We need to find the probability distribution of$X$.

Part (a) Step 2: Explanation

If $H$is heads and $T$is tails, there are $16$different ways to throw the fair coin four times:

Each of these outcomes has an equal chance of occurring$\frac{1}{16}$.

The product of the number of related outcomes and the probability yields the probability distribution of $X=$"the number of heads":

$x$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$\frac{1}{6}$	$\frac{4}{16}$	$\frac{6}{16}$	$\frac{4}{16}$	$\frac{1}{16}$

Part (b) Step 3: Given information 

Given in the question that , you toss a fair coin 4 times. Let X = the number of heads you get

We need to make a histogram of the probability distribution.

Explanation(part b)

Consider the given table:

Each bar's width must be equal, and its height must be equal to the probability:

The histogram is around 2 degrees symmetric.

Part (c) Step 1: Given information

Given in the question that, you toss a fair coin $4$times. Let $X$= the number of heads you get.

We need to find$P(X\le 3)$.

Part (c) Step 2: Explanation

Consider the given table:

Let's add the corresponding probabilities

$$\begin{gathered} P(X\le 3)=P(X=0)+P(X=1)+P(X=2)+P(X=3) \\ =\frac{1}{16}+\frac{4}{16}+\frac{6}{16}+\frac{4}{16} \\ =0.9375 \end{gathered}$$

We should expect to get $3$heard or less in around $93.75$or $94$of the $100$times we perform the quadruple toss of the coin.