Question

There are 3 true coins and 1 false coin with 'head' on both sides. A coin is chosen at random and tossed 4 times. If 'head' occurs all the 4 times, then the probability that the false coin has been chosen and used is

• A. \(\frac{15}{19}\)
• B. \(\frac{14}{19}\)
• C. \(\frac{13}{19}\)
• D. \(\frac{16}{19}\)

Hint:

Stern-Gerlach experiment provides direct evidence of electron spin quantization.

Answer 1

The Correct Option is D

Let A = event that false coin is chosen. P(A) = \(\frac{1}{4}\)
Let B = event of getting 4 heads.
Case 1: False coin chosen: P(B|A) = 1 (since it always shows head)
Case 2: True coin chosen: P(B|A$^c$) = \((\frac{1}{2})^4 = \frac{1}{16}\)
Total probability of B: \[ P(B) = P(A)P(B|A) + P(A^c)P(B|A^c) = \frac{1}{4} \cdot 1 + \frac{3}{4} \cdot \frac{1}{16} = \frac{1}{4} + \frac{3}{64} = \frac{19}{64} \] Using Bayes' Theorem: \[ P(A|B) = \frac{P(A)P(B|A)}{P(B)} = \frac{\frac{1}{4}}{\frac{19}{64}} = \frac{16}{19} \]