Suppose p=0.82. Suppose that the chosen die shows 3. What is the probability that the four-sided die was tossed, i.e., the coin landed on tails?

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We begin by restating the setup and identifying the relevant events. Let T denote the event that the coin landed tails (so the four-sided die was tossed), and let N denote the event that the coin landed heads (so presumably a different die was used). We are told p = 0.82, which will be interpreted as P(T) = 0.82 and therefore P(N) = 1 − P(T) = 0.18. We also have the observed outcome: the chosen die shows a 3.

Next we specify the likelihoods:
- If T occurred (the four-sided die was tossed), the probability that the die shows 3 is 1/4, since a four-sided die has faces 1–4.
- If N occurred (the other die was used, presumably a six-sided die), the probability that the die shows 3 is 1/6, since a standard six-sided die has faces 1–6.

To check the arithmetic more directly with fractions: 0.82 = 82/100 = 41/50, and 0.18 = 9/50. Then
Numerator = (1/4) * (41/50) = 41/200.
Denominator = 41/200 + (1/6) * (9/50) = 41/200 + 9/300 = 41/200 + 3/100 = 41/200 + 6/200 = 47/200.
So P(T | 3) = (41/200) / (47/200) = 41/47, confirming the decimal result.

Summary of the reasoning steps:
- Define T as tails (four-sided die used) with prior P(T) = 0.82, and N as heads (other die) with P(N) = 0.18.
- Use the likelihoods P(3 | T) = 1/4 and P(3 | N) = 1/6.
- Apply Bayes’ rule to compute P(T | 3) = [ (1/4)(0.82) ] / [ (1/4)(0.82) + (1/6)(0.18) ].
- Simplify to 41/47, which equals approximately 0.8723.

COMM_V 190 101-104 2025W1 Final Exam - Practice Questions - Second Half

Suppose p=0.82. Suppose that the chosen die shows 3. What is the probability that the four-sided die was tossed, i.e., the coin landed on tails?

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COMM_V 190 101-104 2025W1 Final Exam - Practice Questions - Second Half

Suppose p=0.82. Suppose that the chosen die shows 3. What is the probability that the four-sided die was tossed, i.e., the coin landed on tails?

View Explanation

Log in for full answers

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A factory produces items on two assembly lines, Line 1 (L1) and Line 2 (L2). 60% of items are produced on L1 and 40% on L2. The defect rate for L1 is 5%, and for L2 is 10%. If a randomly selected item is found to be defective (D), what is the probability that it came from Line 1?