Question

Firefighter glove sizing. Human Factors (December 2015) published a study on how well firefighter gloves fit. In a group of 586 firefighters who reported their glove size, the researchers determined whether the gloves fit well or poorly by gender. The data are summarized in the accompanying table. Consider the gender and glove fit status of a randomly selected firefighter.

a. List the sample points for this experiment.

b. Assign reasonable probabilities to these sample points.

c. Find the probability the firefighter is a female.

d. Find the probability the glove fits well.

e. Find the probability the firefighter is a female and has a well-fitting glove.

f. Find the probability the firefighter is a female or has a well-fitting glove.

	Glove Fits Well	Glove Fits Poorly	Totals
Males	415	132	547
Females	19	50	39
Totals	434	152	586

Source: H. Hsiao, et al., “Firefighter Hand Anthropometry and Structural Glove Sizing: A New Perspective,” Human Factors, Vol. 57, No. 8, December 2015 (Table 6).

Short answer

1. S = [MF, MD, FF, FD]
2. 0.7082, 0.2253, 0.0324, 0.0341
3. 0.0666
4. 0.7406
5. 0.0324
6. 0.7747

Step-by-step solution

Identify the sample points

The statistical tool of probability is used to investigate unpredictability. It is concerned with the likelihood of an event occurring. In this experiment, the sample points are as follows:

S = [MF, MD, FF, FD]

MF Indicates Males for whom the glove fits well.

MD Indicates Males for whom the glove fits poorly.

FF Indicates Females for whom the glove fits well.

FD Indicates Females for whom the glove fits poorly.

 Allocate appropriate probability to these sample points

The probabilities are derived as follows:

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{MF}\mathbf{\right)}\mathbf{=}\frac{\mathbf{415}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{7082} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{MD}\mathbf{\right)}\mathbf{=}\frac{\mathbf{132}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2253} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{FF}\mathbf{\right)}\mathbf{=}\frac{\mathbf{19}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0324} \end{gathered}$$

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{FD}\mathbf{\right)}\mathbf{=}\frac{\mathbf{20}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0341} \end{gathered}$$

Determine the probability that the firefighter is female

$$\begin{gathered} \mathbf{P}\mathbf{\left(}\mathbf{Female}\mathbf{\right)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{FF}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{FD}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{19}}{\mathbf{586}}\mathbf{+}\frac{\mathbf{20}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{39}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0666} \end{gathered}$$

Determine the probability that the glove fits well

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{glove}\mathbf{}\mathbf{fits}\mathbf{}\mathbf{well}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{\left(}\mathbf{MF}\mathbf{\right)}\mathbf{+}\mathbf{P}\mathbf{\left(}\mathbf{FF}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{415}}{\mathbf{586}}\mathbf{+}\frac{\mathbf{19}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{434}}{\mathbf{586}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{7406} \end{gathered}$$

Determine the probability that the firefighter is a female and has a well-fitting glove

$\begin{array}{c}\mathbf{P}\mathbf{\left(}\mathbf{FF}\mathbf{\right)}\mathbf{=}\frac{\mathbf{19}}{\mathbf{586}}\\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0324}\end{array}$

Determine the probability that the firefighter is a female or has a well-fitting glove

$$\begin{gathered} \mathbf{P}\mathbf{(}\mathbf{female}\mathbf{}\mathbf{or}\mathbf{}\mathbf{glove}\mathbf{}\mathbf{fits}\mathbf{}\mathbf{well}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{–}\mathbf{P}\mathbf{(}\mathbf{male}\mathbf{}\mathbf{and}\mathbf{}\mathbf{glove}\mathbf{}\mathbf{fits}\mathbf{}\mathbf{poorly}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{–}\mathbf{0}\mathbf{.}\mathbf{2253} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{7747} \end{gathered}$$