Question

In a random sample of 106 social (or service) robots designed to entertain, educate, and care for human users, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. One of the 106 social robots is randomly selected and the design (e.g., wheels only) is noted.

1. List the sample points for this study.
2. Assign reasonable probabilities to the sample points.
3. What is the probability that the selected robot is designed with wheels?
4. What is the probability that the selected robot is designed with legs?

Short answer

1. There are four sample points which are as follows:$\left\{\mathbf{L}\mathbf{,}\mathbf{W}\mathbf{,}\mathbf{LW}\mathbf{,}\mathbf{N}\right\}$
2. The probability of respective sample points is-$\mathbf{P}\mathbf{\left(}\mathbf{L}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5943}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{W}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1886}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{LW}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0754}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{N}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1415}$
3. The probability that the selected robot is designed with wheels is 0.264
4. Theprobability thatthe selected robot is designed with legs is 0.6697

Step-by-step solution

Defining the events and listing the sample points in mathematical format

As per the information given in the question, one notes down the events and assigns them initials. Let’s assign-

L = Legs only

W = Wheels only

LW = Both legs and wheels

N = Neither legs nor wheels

The sample points are – $\left\{\mathbf{L}\mathbf{,}\mathbf{W}\mathbf{,}\mathbf{LW}\mathbf{,}\mathbf{N}\right\}$

Probability is defined as the total number of favorable outcomes from the total number of given outcomes.

Finding the probability of selecting a robot designed with legs only, wheels only, legs and wheels

Total number of robots designed with legs only = 63

Total number of robots = 106

$$\begin{gathered} \mathbf{Probability}\mathbf{}\mathbf{of}\mathbf{}\mathbf{selecting}\mathbf{}\mathbf{a}\mathbf{}\mathbf{root}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{legs}\mathbf{}\mathbf{only} \\ \mathbf{=}\frac{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{legs}\mathbf{}\mathbf{only}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}} \end{gathered}$$

That is, the probability of selecting a robot designed with legs only $$\begin{gathered} \mathbf{=}\frac{\mathbf{63}}{\mathbf{106}} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5943} \end{gathered}$$

Total number of robots with wheels only = 20

Total number of robots = 106

localid="1662213507997" $$\begin{gathered} \mathbf{Probability}\mathbf{}\mathbf{of}\mathbf{}\mathbf{selecting}\mathbf{}\mathbf{a}\mathbf{}\mathbf{robot}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{wheels}\mathbf{}\mathbf{only} \\ \mathbf{=}\frac{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{wheels}\mathbf{}\mathbf{only}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}} \end{gathered}$$

Probability of selecting a robot designed with wheels only $\mathbf{=}\frac{\mathbf{20}}{\mathbf{106}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1886}$

Total number of robots designed with legs and wheels only = 8

Total number of robots = 106

localid="1662213573058" $$\begin{gathered} \mathbf{Probability}\mathbf{}\mathbf{of}\mathbf{}\mathbf{selecting}\mathbf{}\mathbf{a}\mathbf{}\mathbf{robot}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{legs}\mathbf{}\mathbf{and}\mathbf{}\mathbf{wheels} \\ \mathbf{=}\frac{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{legs}\mathbf{}\mathbf{and}\mathbf{}\mathbf{wheels}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}} \end{gathered}$$

Probability of selecting a robot designed with legs and wheels $\mathbf{=}\frac{\mathbf{8}}{\mathbf{106}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0754}$

Obtaining probability of selecting a robot designed with neither legs nor wheels

Total number of robots designed with neither legs nor wheels = 15

Total number of robots = 106

$$\begin{gathered} \mathbf{Probability}\mathbf{}\mathbf{of}\mathbf{}\mathbf{selecting}\mathbf{}\mathbf{a}\mathbf{}\mathbf{robot}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{neither}\mathbf{}\mathbf{legs}\mathbf{}\mathbf{nor}\mathbf{}\mathbf{wheels} \\ \mathbf{=}\frac{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}\mathbf{}\mathbf{designed}\mathbf{}\mathbf{with}\mathbf{}\mathbf{neither}\mathbf{}\mathbf{legs}\mathbf{}\mathbf{nor}\mathbf{}\mathbf{wheels}}{\mathbf{Total}\mathbf{}\mathbf{number}\mathbf{}\mathbf{of}\mathbf{}\mathbf{robots}} \end{gathered}$$

Probability of selecting a robot designed with neither legs nor wheels $\mathbf{=}\frac{\mathbf{15}}{\mathbf{106}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1415}$

Assigning reasonable probabilities to sample points

The probabilities calculated in steps 1 to 4 can be assigned as $\mathbf{P}\mathbf{\left(}\mathbf{L}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5943}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{W}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1886}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{LW}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0754}\mathbf{,}\mathbf{P}\mathbf{\left(}\mathbf{N}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1415}$ respectively.

Noting down the probability of selecting a robot designed with wheels only and legs and wheels

As we have already calculated it earlier in part (b), Step 2, one notes the probability of selecting a robot designed with wheels only to move further.

We calculated it earlier in part (b), Step 3, which notes down the probability of selecting a robot with legs and wheels to move further.

Detecting probability of selecting a robot that is designed with wheels

The probability that the selected robot is designed with wheels is given by the summation of the probability of selecting a robot designed with wheels only and the probability of selecting a robot designed with legs and wheels, that is,

$\mathbf{P}\mathbf{\left(}\mathbf{W}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{LW}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1886}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{0754}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2640}$

Put down the probability of selecting a robot designed with legs only and legs and wheels

One calculates it earlier in part (b), Step 1, one notes down the probability of selecting a robot designed with legs only as $\mathbf{P}\mathbf{\left(}\mathbf{L}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5943}$ to move further.

One calculated it earlier in part (b), Step 3, one notes down the probability of selecting a robot designed with legs and wheels as $\mathbf{P}\mathbf{\left(}\mathbf{LW}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0754}$ to move further.

Realizing probability of selecting a robot that is designed with legs

The probability that the selected robot is designed with legs is given by the summation of the probability of selecting a robot with legs only and the probability of selecting a robot with legs and wheels

$\mathbf{P}\mathbf{\left(}\mathbf{L}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{LW}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5943}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{754}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{6697}$