Question

For a biased die the probabilities for different faces to turn up are given below. The die is tossed and you are told that either face 1 or 2 has turned up. What is the probability of getting face 1?

Face	1	2	3	4	5	6
Probability	0.1	0.32	0.21	0.15	0.05	0.17

A. \[\dfrac{5}{{21}}\]
B. \[\dfrac{5}{{22}}\]
C. \[\dfrac{4}{{21}}\]
D. None of these

Answer 1

Hint:First we will calculate the probability of getting face 1 or 2 by using the formula \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)\]. Then we will apply the formula \[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\].

Formula Used:
\[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)\]
\[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\]

Complete step by step solution:
The probability of getting face 1 is 0.1.
The probability of getting face 2 is 0.32.
Since face 1 and face 2 are independent events.
Apply the formula \[P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)\] to get face 1 or face 2
The probability of getting face 1 or face 2 is \[0.1 + 0.32 = 0.42\].
Apply the formula \[{\rm{Probability = }}\dfrac{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{favorable}}\,{\rm{outcomes}}}}{{{\rm{The}}\,{\rm{number}}\,{\rm{of}}\,{\rm{total}}\,{\rm{outcomes}}}}\] to calculate the probability of getting face 1.
A bise die rolled. After rolling the die, we will get either face 1 or face 2. So, the total probability is the probability of getting either face 1 or face 2.
The number of total outcomes is 0.42.
The number of favorable outcomes is 0.1.
The required probability is \[{\rm{ = }}\dfrac{{0.1}}{{0.42}}\]
Multiply 100 with numerator and denominator
                                              \[{\rm{ = }}\dfrac{{0.1 \times 100}}{{0.42 \times 100}}\]
                                             \[{\rm{ = }}\dfrac{{10}}{{42}}\]
Divide the numerator and denominator by 2
                                          \[{\rm{ = }}\dfrac{5}{{21}}\]
Hence option A is the correct.

Note:Student often do a common mistake to calculate total probability. They calculate the probability by adding the probability of getting face 1, face 2, face 3, face 4, face 5, and face 6. Since the die is biased, so it shows only face 1 and face 2 when it rolled. Therefore, the total probability is the sum of the probability of getting face 1 and face 2.