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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.

Denote events

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

We are looking for the probability that sold shirt is a medium, long-sleeved, print shirt and we can find that in the table 鈥� 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),

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

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).

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 \({\rm{SS}}\)

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

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.

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

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.

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.