Question

Tamara is reviewing recent orders at her deli to determine which meats she should order. She found that of 1,000 orders, 450 customers ordered turkey, 375 customers ordered ham, and 250 customers ordered neither turkey nor ham. Based on this data, how many of the next 1,000 customers will order both turkey and ham? Show your work, and use complete sentences.

Answer 1

Let
\(T\)=turkey
\(H\)=ham

You are given:
\(P(T)=450/1000 = 0.45\)
\(P(H)=375/1000=0.375\)
\(P(\overline{H} \cap \overline{T})=250/1000=0.25\)

You need:
\(P(H \cap T)\)

So:
From deMorgan's law, \(P(\overline{H \cup T})=P(\overline{H}\cap\overline{T})\)

By complements, \(P(\overline{H \cup T})=1-P(H \cup T) \\ \implies P(H \cup T)=1-P(\overline{H \cup T})=1-P(\overline{H}\cap\overline{T})\)

By the rule for unions:
\(P(H \cup T) = P(H)+P(T)-P(H\cap T)\) ... which is the same as
\(1-P(\overline{H}\cap\overline{T})=P(H)+P(T)-P(H \cap T) \\ \implies P(H \cap T)=P(H)+P(T)-1+P(\overline{H}\cap\overline{T})\)