Question

Mark the correct alternative in each of the following:

Three biased coins were tossed 800 times simultaneously. The outcomes are given in the following table :

Outcome : No head One Head Two Head

Frequency : 120 280 x

If the probability of occurrence of two heads is thrice that of all heads, then the value of x.
(a) 150
(b) 240
(c) 300
(d) 340

Answer 1

Three biased coins are tossed 800 times.

∴ Total number of trials = 800

Let E be the event of occurrence of two heads and F be the event of occurrence of all heads.

∴ Probability of occurrence of two heads = P(E) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{trials}\mathrm{of}\mathrm{occurrence}\mathrm{of}\mathrm{two}\mathrm{heads}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{x}{800}$ .....(1)

Now,

Number of trials of occurrence of all heads (or three heads)

= Total number of trials − Number of trials of occurrence of no head − Number of trials of occurrence of one head − Number of trials of occurrence of two heads

= 800 − 120 − 280 − x

= 400 − x

∴ Probability of occurrence of all heads = P(F) = $\frac{\mathrm{Number}\mathrm{of}\mathrm{trials}\mathrm{of}\mathrm{occurrence}\mathrm{of}\mathrm{all}\mathrm{heads}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{trials}}=\frac{400-x}{800}$ .....(2)

It is given that,

Probability of occurrence of two heads = 3 × Probability of occurrence of all heads

$\therefore \frac{x}{800}=3\times \left(\frac{400-x}{800}\right)$ [From (1) and (2)]

$\Rightarrow x=1200-3x$

$\Rightarrow 4x=1200$

$\Rightarrow x=\frac{1200}{4}=300$

Thus, the value of x is 300.

Hence, the correct answer is option (c).