Question

Find the probability of getting $2$or$3$ tails when a coin is tossed four times.
\[
  (a)\,\,\,\dfrac{7}{{16}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
  (b)\,\,\dfrac{9}{{16}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
  (c)\,\,\dfrac{5}{8}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
  (d)\,\,\dfrac{5}{{16}} \\
 \]

Answer 1

Hint: Use binomial distribution and taking \[n = 4,\,\,\,\,\,r = 2\,\,\,\,or\,\,r = 3\,\,\]to get the result. Probability of an event is:
\[P\left( E \right) = \dfrac{{favourable{\text{ }}outcomes}}{{total\,\,number\,\,of\,\,outcomes}}\].

Complete step by step answer:
(1) When a coin is tossed, probability of getting a head is\[ = \dfrac{1}{2}\,\]
Probability of getting a tail when a coin is tossed\[ = \dfrac{1}{2}\]
\[\therefore \,\,p = \dfrac{1}{2},\,\,q = \dfrac{1}{2}\]
(2) Coin is tossed$4$times
\[\therefore \,\,n = 4\]
(3) Probability of getting $2$or$3$tails,
\[r = 3\,\,or\,\,\,r = 2\]
(4) Using formula of binomial distribution\[^n{C_r}\,{(p)^r}\,{(q)^{n - r}}\,\]
Here, n is the total number of ways.
r is required.
p is success in one case
q is failure in one case
(5) Here, \[n = 4,\,\,r = 2\,\,or\,\,r = 3,\,\,p = \dfrac{1}{2},\,\,q = \dfrac{1}{2}\]
(6) Using values in formula mentioned in step (4), we get
\[^4{C_2}{\left( {\dfrac{1}{2}} \right)^2}{\left( {\dfrac{1}{2}} \right)^2} + {\,^4}{C_3}{\left( {\dfrac{1}{2}} \right)^3}{\left( {\dfrac{1}{2}} \right)^1}\]
\[ = \dfrac{{4!}}{{2!\,2!}} \times \dfrac{1}{4} \times \dfrac{1}{4} + \dfrac{{4!}}{{3!\,1!}} \times \dfrac{1}{8} \times \dfrac{1}{2}\]
$ = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} \times \dfrac{1}{4} \times \dfrac{1}{4} + \dfrac{{4 \times 3!}}{{3! \times 1}} \times \dfrac{1}{8} \times \dfrac{1}{2}$
\[ = \dfrac{{4 \times 3}}{2} \times \dfrac{1}{{16}} + 4 \times \dfrac{1}{8} \times \dfrac{1}{2}\]
\[ = \dfrac{3}{8} + \dfrac{1}{4}\]
\[ = \dfrac{{3 + 2}}{8}\]
\[ = \dfrac{5}{8}\]
There probability of getting two or three tails if the coin is tossed $4$ times is \[\dfrac{5}{8}\]
Hence, the correct option is \[(c)\,\,\left( {\dfrac{5}{8}} \right)\]

Additional Information: Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event. The value is expressed from zero to one. Probability has been introduced in Maths to predict how likely events are to happen.

Note: When an object is tossed n times, we use binomial distribution in probability.