Question

Toss two fair coins and let X equal the number of heads observed .Find the probability distribution for X.

Answer 1

If X = number of heads after two fair coins are tossed then, X can take the values $0,1and 2$

Since tossing the coins two times are independent from each other,

Probability of getting head or tails on first toss or second toss will be equal to $1/2=0.5$

$1) X=0$ is possible only in one case if tails comes in both the coins. So,

Possible cases: $TT$

Where T denotes tails, TT means tails on first toss and tails on second toss.

$P(X=0)= P(TT)$

where $P(TT)$ means probability that Tails comes on both the coins.

$P(TT)= 0.5*0.5=0.25$

$2) X=1$

Possible cases : $HT,TH$

HT means heads on first toss and tails on second toss, TH m and tails on first toss and heads on second toss.

$P(X=1)=P(HT)+P(TH)$

$P(HT) = 0.5*0.5=0.25$

$P(TH)=0.5*0.5=0.25$

$P(X=1)=0.25+0.25=0.50$

$3) X=2$

Possible cases: $HH$

Where H denotes heads, HH means heads on first toss and heads on second toss.

$P(X=2)=P(HH)$

$P(HH)=0.5*0.5=0.25$

So, $P(X=2)=0.25$

Now, we can write the probability distribution of X as:

X P(X)

0 0.25

1 0.5

2 0.25

We can also cross check the distribution by summing all the probability, that should be equal to 1.