Question

a study was conducted of the trend in the design of social robots. in a random sample 105 social robots, 62 were built with legs only, 19 were built with wheels only, 10 were built with both legs and wheels, and 14 were built with neither legs nor wheels. Use the rule of complements to find the probability that a randomly selected social robot is designed with wheels or without legs.

Answer 1

Solution :

Let us define some events as follows :

A : A randomly selected robot was built with legs only

B : A randomly selected robot was built with wheels only

A ∩ B : A randomly selected robot was built with both legs and wheels

A​​​​​​c ∩ B​​​​​​c : A randomly selected robot was built with neither legs nor wheels.

We have following informations :

n(A) = 62, n(B) = 19, n(A ∩ B) = 10, n(A​​​​​​c ∩ B​​​​​​c) = 14

total number of robots n(S) = 105

Where, n(A) represents the number of outcomes which are favourable to event A.

Similarly, n(B) = 19, n(A ∩ B) = 10, n(A​​​​​​c ∩ B​​​​​​c) = 14 represents their respective numbers.

Probability of an event A is given as follows :

P(A) = 62/105, P(B) = 19/105, P(A ∩ B) = 10/105 and P(A​​​​​​c ∩ B​​​​​​c) = 14/105

We have to obtain P(B ∪ A​​​​​​c).

P(B  ∪ A​​​​​​c) = P(B) + P(A​​​​​​c) - P(B ∩ A​​​​​​c)

[Since, P(A​​​​​​c) = 1 - P(A) and P(B ∩ A​​​​​​c) = P(B) - P(A ∩ B)]

Hence,

P(B  ∪ A​​​​​​c) = P(B) + 1 - P(A​​​​​) - [P(B) - P(A ∩ B)]

P(B  ∪ A​​​​​​c) = P(B) + 1 - P(A​​​​​) - P(B) + P(A ∩ B)

P(B  ∪ A​​​​​​c) = 1 - P(A​​​​​) + P(A ∩ B)

P(B  ∪ A​​​​​​c) = 1 - (62/105) + (10/105)

P(B  ∪ A​​​​​​c) = 1 - (52/105)

P(B  ∪ A​​​​​​c) = (105 - 52)/105

P(B  ∪ A​​​​​​c) = 53/105

Hence, the probability that a randomly selected social robot is designed with wheels or without legs is 53/105.

Answer 2

The rule of complements states that the probability of an event happening is equal to 1 minus the probability of the event not happening.

Let A be the event that a social robot is designed with wheels, and B be the event that a social robot is designed without legs. We want to find P(A or B), which is the probability that a social robot is designed with wheels or without legs.

We can find P(A) and P(B) using the given information:

• P(A) = (19 + 10) / 105 = 0.2762 (19 robots have wheels only, and 10 have both legs and wheels, so there are 29 robots with wheels)

• P(B) = (19 + 14) / 105 = 0.2762 (19 robots have wheels only, and 14 have neither legs nor wheels, so there are 72 robots with legs or both legs and wheels)

To find P(A or B), we can use the rule of complements:

P(A or B) = 1 - P(not A and not B)

The probability of not A and not B can be found by subtracting the probabilities of A and B from 1:

P(not A and not B) = 1 - P(A) - P(B) = 1 - 0.2762 - 0.2762 = 0.4476

Therefore,

P(A or B) = 1 - P(not A and not B) = 1 - 0.4476 = 0.5524

So the probability that a randomly selected social robot is designed with wheels or without legs is 0.5524, or about 55.24%.