Question

A box contains three coins, one coin is fair, one coin is two headed and one coin is weighted so that the probability of head appearing is \[\dfrac{1}{3}\]. A coin is selected at random and tossed. Find the probability that the head appears.

Answer 1

Hint: First, we will find the probability of getting a fair coin, two headed coin and weighted coin from the box. Then, we will find the probability of getting heads out of the fair coin, two headed coin and weighted coin. Then, we will multiply the probability of getting a fair coin, two headed coin and weighted coin with the probability of getting head from the fair coin, two headed coin and weighted coin respectively. Then, we will add these probabilities.

Complete step-by-step answer:
Given that there are three coins, one is a fair coin.
Second one is a two headed coin.
And the third one is a weighted coin.
Out of the three coins,
The probability of getting the fair coin is \[\dfrac{1}{3}\].
After tossing the fair coin,
The probability of getting a head out of a fair coin is \[\dfrac{1}{2}\].
The probability of getting a two headed coin is \[\dfrac{1}{3}\].
After tossing the two headed coin,
The probability of getting a head out of the two headed coin is 1.
The probability of getting a weighted coin is \[\dfrac{1}{3}\].
Given that the probability of getting head out of a weighted coin is \[\dfrac{1}{3}\].
Therefore, the probability of getting head is
 \[ = \dfrac{1}{3} \times \dfrac{1}{2} + \dfrac{1}{3} \times 1 + \dfrac{1}{3} \times \dfrac{1}{3}\]
On simplification, we get
 \[ = \dfrac{1}{6} + \dfrac{1}{3} + \dfrac{1}{9}\]
Solve by taking 18 as L.C.M.
 \[ = \dfrac{{3 + 6 + 2}}{{18}}\]
On simplification we get,
 \[ = \dfrac{{11}}{{18}}\]
Therefore, the probability of getting a head is \[\dfrac{{11}}{{18}}\].

Note: Try to remember these basic formula to solve these types of questions:-
The probability of getting a coin out of three coins is \[\dfrac{1}{3}\]
The probability of getting a head out of a fair coin is $\dfrac{1}{2}$.
The probability of getting a head out of a two headed coin is 1.