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TIFR CSE 2013 | Part A | Question: 4

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A biased coin is tossed repeatedly. Assume that the outcomes of different tosses are independent and probability of heads is $\dfrac{2}{3}$ in each toss. What is the probability of obtaining an even number of heads in $5$ tosses, zero being treated as an even number?

  1. $\left(\dfrac{121}{243}\right)$
  2. $\left(\dfrac{122}{243}\right)$
  3. $\left(\dfrac{124}{243}\right)$
  4. $\left(\dfrac{125}{243}\right)$
  5. $\left(\dfrac{128}{243}\right)$
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probability of obtaining an even number of heads in $5$ tosses, zero being treated as an even number 
number of event = $0$ head or $2$ head or $4$ head 

Probability of head =$ \dfrac{2}{3}$
Probability of tail = $\dfrac{1}{3}$

Probability = ${^5}C{_0}\left(\dfrac{2}{3}\right)^{0}\left(\dfrac{1}{3}\right)^{5}+{^5}C{_2}\left(\dfrac{2}{3}\right)^{2}\left(\dfrac{1}{3}\right)^{3} +{^5}C{_4}\left(\dfrac{2}{3}\right)^{4}\left(\dfrac{1}{3}\right)^{1}$
                
                    =$\dfrac{121}{243}.$ 

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