A box contains 100 bolts and 50 nuts. It is given that 50% bolts and 50% nuts are rusted. Two objects are selected from the box at random. Find the probability that either both are rusted.

Open in App

Solution

Total number of objects = (100 + 50) = 150.

Let S be the sample space. Then,

n(S) = number of ways of selecting 2 objects out of 150 $=^{150} C_{2} \cdot$

Number of rusted objects

= (50% of 100) + (50% of 50) = (50 + 25) = 75.

Let $E_{1}$ = event of selecting 2 bolts out of 100 bolts,

and $E_{2}$ = event of selecting 2 rusted objects out of 75 rusted objects

$∴ (E_{1} \cap E_{2})$ = event of selecting 2 rusted bolts out of 50 rusted bolts

$∴ n (E_{1})$ = number of ways os selecting 2 bolts out of 100 $=^{100} C_{2} \cdot$

$∴ n (E_{2})$ = number of ways os selecting 2 rusted objects out of $75 =^{75} C_{2} \cdot$

$∴ P (E_{1}) = \frac{n (E_{1})}{n (S)} = \frac{^{100} C_{2}}{^{150} C_{2}}, P (E_{2}) = \frac{n (E_{2})}{n (S)} = \frac{^{75} C_{2}}{^{150} C_{2}}$

and $P (E_{1} \cap E_{2}) = \frac{n (E_{1} \cap E_{2})}{n (S)} = \frac{^{50} C_{2}}{^{150} C_{2}} .$

P(Selecting both bolts or both rusted objects)

$= P (E_{1} o r E_{2}) = P (E_{1} \cup E_{2})$

$= P (E_{1}) + P (E_{2}) - P (E_{1} \cap E_{2})$

$= \frac{^{100} C_{2}}{^{150} C_{2}} + \frac{^{75} C_{2}}{^{150} C_{2}} - \frac{^{50} C_{2}}{^{150} C_{2}} = \frac{(^{100} C_{2} +^{75} C_{2} -^{50} C_{2})}{^{150} C_{2}}$

$= \frac{(4950 + 2775 - 1225)}{11175} = \frac{6500}{11175} = \frac{260}{447} = 0.58.$

Hence, the required probability is 0.58.

Suggest Corrections

Similar questions

Q. 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.

Q. A box contains 30 bolts and 40 nuts. Half of the bolts and half of the nuts are rusted. If two items are drawn at random from the box, what is the probability that either both are rusted or both are bolts?

Q. A drawer contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or a bolt ?

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is

Q. A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is

(a)

\frac{3}{16}

(b)

\frac{5}{16}

(c)

\frac{11}{16}

(d)

\frac{14}{16}

Join BYJU'S Learning Program