A boy contains $100 b o l t s$ and $50 n u t s$ . It is given that $50 % b o l t s$ and $50 % n u t s$ are rusted. Two objects are selected from the box at random. Find the probability that both are bolts or both are rusted.

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Solution

Let A = Event of getting rusted item

B = Event of getting a bolt

$A ⋂ B$ = Event of getting a bolt which is rusted

Total no of items = 150

Total rusted items = 0.5(100)+0.5(50) = 75

$P (A) = \frac{^{75} C_{2}}{^{100} C_{2}} = \frac{75 \times 74}{150 \times 149} = \frac{37}{149}$

$P (B) = \frac{^{100} C_{2}}{^{150} C_{2}} = \frac{100 \times 99}{150 \times 149} = \frac{66}{149}$

$P (A ⋂ B) = \frac{^{50} C_{2}}{^{150} C_{2}} = \frac{50 \times 49}{150 \times 149} = \frac{49}{447}$

P (Getting both bolts on both rusted) = $P (A) + P (B) - P (A ⋂ B)$ $= \frac{260}{447} = 0.5816$

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