Question

A department store sells sports shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.

a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?

b. What is the probability that the next shirt sold is a medium print shirt?

c. What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt?

d. What is the probability that the size of the next shirt sold is the medium? That the pattern of the next shirt sold as a print?

e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?

f. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved?

Short answer

a. The probability that the next shirt sold is a medium, long-sleeved, print shirt is 0.05.

b. The probability that the next shirt sold is a medium print shirt is 0.12.

c. The probability that the next shirt sold is short-sleeved is 0.56 & A long-sleeved shirt is 0.44.

d. The probability that the size of the next shirt sold is the medium is 0.25.

e. The probability that its size was medium is 0.533.

f. The probability that it was short-sleeved is 0.566.

Step-by-step solution

Definition of Probability

Probability is a discipline of mathematics concerned with numerical explanations of the likelihood of an event occurring or the truth of a claim. The probability of an event is a number between \(0{\rm{ }} and {\rm{ }}1\), with zero indicating impossibility and one indicating certainty, broadly speaking.

Given Data

Denote events

\(\begin{array}{l}M = \{ {\rm{ sold shirt is a medium }}\} \\LS = \{ {\rm{ sold shirt is a long - sleeved shirt }}\} ;\\SS = \{ {\rm{ sold shirt is a short - sleeved shirt }}\} ;\\Pr = \{ {\rm{ sold shirt is a print shirt }}\} ;\\Pl = \{ {\rm{ sold shirt is a plaid shirt }}\} \end{array}\)

Calculation for determining probability in part a.

We are looking for the probability that sold shirt is a medium, long-sleeved, print shirt and we can find that in the table – first, we look for a Long-sleeved table, in the table we find medium size (which is "M" row) and a print shirt (which is "Pr" column),

\(P(M,LS,P) = 0.05.{\rm{ }}\)

Calculation for determining probability in part b.

We are looking for the probability that sold shirt is a medium, print shirt. It can either be a short-sleeved (SS) or long-sleeved (LS) shirt so

\(\begin{array}{c}P(M,Pr) &=& P(M,Pr,LS) + P(M,Pr,SS)\\ &=& 0.05 + 0.07\\ &=& 0.12\end{array}\)

where we find the probabilities in the table as described in (a).

Calculation for determining probability in part c.

We are looking for the probability that sold shirt is a short-sleeved shirt. There are nine different combinations (all combinations in the short-sleeved table), it can be plaid, print, and stripe and also small, medium, or large, which makes it \(3 \times 3 = 9\)combinations

\(\begin{array}{c}P(SS) &=& {\rm{ sum of probabilities of all }}9{\rm{ combinations }}\\ &=& 0.04 + 0.02 + \ldots + 0.07 + 0.08\\ &=& 0.56\end{array}\)

where we find the probabilities in the table as described in (a).

The probability that sold shirt is a short-sleeved shirt can be calculated using \(P(A) + P\left( {{A^\prime }} \right) = 1\), where the complement of the event LS is \({\rm{SS}}\)

\(\begin{array}{c}P(LS) &=& 1 - P(SS)\\ &=& 1 - 0.56\\ &=& 0.44.\end{array}\)

Calculation for determining probability in part d.

The probability that the sold garment is a medium shirt is what we're looking for. There are six possible combinations for medium shirts: first, there are plaid/print/stripe shirts in a short-sleeved table, and second, there are plaid/print/stripe shirts in a long-sleeved table, for a total of six choices. The total of the six values from the tables is the likelihood.

\(P(M) = 0.08 + 0.07 + 0.12 + 0.10 + 0.05 + 0.07 = 0.49.\)

We're looking for a chance that the shirt was sold as a print shirt. There are six different print shirt combinations: first, there are small/medium/large shirts in a short-sleeved table, and second, there are small/medium/large shirts in a long-sleeved table, for a total of six options. The total of the six values from the tables is the likelihood.

\(P(Pr) = 0.02 + 0.07 + 0.07 + 0.02 + 0.05 + 0.02 = 0.25\)

Calculation for determining probability in part e.

Given that the shirt just sold was a short-sleeved plaid, the probability that its size was medium, is a conditional probability of M given that the event \(SS \cap Pl\)(Pl is plaid) has occurred.

The conditional probability of A given that the event B has occurred, for which \(P\left( B \right) > 0,\)is

\(P(A\mid B) = \frac{{P(A \cap B)}}{{P(B)}}\)

for any two events A and B.

From the definition, we have

\(P(M\mid SS \cap Pl) = \frac{{P(M \cap SS \cap Pl)}}{{P(SS \cap Pl}}\mathop = \limits^{(1)} \frac{{0.08}}{{0.04 + 0.08 + 0.03}} = 0.533\)

(1): we use the same reasoning as we did previously.

Calculation for determining probability in part f

Given that the shirt just sold was a medium plaid, the probability that it was short-sleeved, is the conditional probability of SS given that the event \(M \cap PI\)has occurred.

From the definition, we have

\(P(SS\mid M \cap Pl){\rm{ }} = \frac{{P(SS \cap M \cap Pl)}}{{P(M \cap Pl}}\mathop = \limits^{(1)} \frac{{0.08}}{{0.08 + 0.1}} = 0.444\)

(1): we use the same reasoning as we did previously.

For long-sleeved, we use \(P(A) + P\left( {{A^\prime }} \right) = 1\), where the complement is the probability we have just calculated

\(\begin{array}{c}P(LS\mid M \cap Pl) &=& 1 - P(SS\mid M \cap Pl)\\ &=& 1 - 0.444\\ &=& 0.556.\end{array}\)

You could have calculated the probability the same way as we did for LS.