Answer on Question #53170 – Math – Statistics and Probability
A,B,C,D cut a pack of 52 cards successively in the order given. If a person who cuts a spade first receives rs.700, what are their respective expectations?
Solution
The probability of the first spade for the first person is
P ( A ) = 13 52 = 0.25. P(A) = \frac{13}{52} = 0.25. P ( A ) = 52 13 = 0.25.
The respective expectation for the first person is
E ( A ) = x ⋅ P ( x ) = 700 ⋅ 0.25 = 175. E(A) = x \cdot P(x) = 700 \cdot 0.25 = 175. E ( A ) = x ⋅ P ( x ) = 700 ⋅ 0.25 = 175.
The probability of the first spade for the second person is
P ( B ) = P ( B ∣ A ‾ ) ⋅ P ( A ‾ ) = 13 51 ⋅ ( 1 − 13 52 ) = 13 ⋅ ( 52 − 13 ) 51 ⋅ 52 = 39 51 ⋅ 4 = 13 17 ⋅ 4 . P(B) = P(B|\overline{A}) \cdot P(\overline{A}) = \frac{13}{51} \cdot \left(1 - \frac{13}{52}\right) = \frac{13 \cdot (52 - 13)}{51 \cdot 52} = \frac{39}{51 \cdot 4} = \frac{13}{17 \cdot 4}. P ( B ) = P ( B ∣ A ) ⋅ P ( A ) = 51 13 ⋅ ( 1 − 52 13 ) = 51 ⋅ 52 13 ⋅ ( 52 − 13 ) = 51 ⋅ 4 39 = 17 ⋅ 4 13 .
The respective expectation for the second person is
E ( B ) = x ⋅ P ( B ) = 700 ⋅ 13 17 ⋅ 4 ≈ 133.82. E(B) = x \cdot P(B) = 700 \cdot \frac{13}{17 \cdot 4} \approx 133.82. E ( B ) = x ⋅ P ( B ) = 700 ⋅ 17 ⋅ 4 13 ≈ 133.82.
The probability of the first spade for the third person is
P ( C ) = P ( B ∣ A ‾ , B ‾ ) ⋅ P ( A ‾ , B ‾ ) = 13 50 ⋅ 52 − 13 52 ⋅ 51 − 13 51 = 13 50 ⋅ 39 52 ⋅ 38 51 = 13 ⋅ 38 50 ⋅ 4 ⋅ 17 = 13 ⋅ 19 17 ⋅ 100 . P(C) = P(B|\overline{A}, \overline{B}) \cdot P(\overline{A}, \overline{B}) = \frac{13}{50} \cdot \frac{52 - 13}{52} \cdot \frac{51 - 13}{51} = \frac{13}{50} \cdot \frac{39}{52} \cdot \frac{38}{51} = \frac{13 \cdot 38}{50 \cdot 4 \cdot 17} = \frac{13 \cdot 19}{17 \cdot 100}. P ( C ) = P ( B ∣ A , B ) ⋅ P ( A , B ) = 50 13 ⋅ 52 52 − 13 ⋅ 51 51 − 13 = 50 13 ⋅ 52 39 ⋅ 51 38 = 50 ⋅ 4 ⋅ 17 13 ⋅ 38 = 17 ⋅ 100 13 ⋅ 19 .
The respective expectation for the third person is
E ( C ) = x ⋅ P ( B ) = 700 ⋅ 13 ⋅ 19 17 ⋅ 100 ≈ 101.71. E(C) = x \cdot P(B) = 700 \cdot \frac{13 \cdot 19}{17 \cdot 100} \approx 101.71. E ( C ) = x ⋅ P ( B ) = 700 ⋅ 17 ⋅ 100 13 ⋅ 19 ≈ 101.71.
The probability of the first spade for the fourth person is
P ( D ) = P ( B ∣ A ‾ , B ‾ , C ‾ ) ⋅ P ( A ‾ , B ‾ , C ‾ ) = 13 49 ⋅ 52 − 13 52 ⋅ 51 − 13 51 ⋅ 50 − 13 50 = 13 49 ⋅ 39 52 ⋅ 38 51 ⋅ 37 50 = 13 49 ⋅ 1 4 ⋅ 38 17 ⋅ 37 50 = = 13 49 ⋅ 1 2 ⋅ 19 17 ⋅ 37 50 . \begin{array}{l}
P(D) = P(B|\overline{A}, \overline{B}, \overline{C}) \cdot P(\overline{A}, \overline{B}, \overline{C}) = \frac{13}{49} \cdot \frac{52 - 13}{52} \cdot \frac{51 - 13}{51} \cdot \frac{50 - 13}{50} = \frac{13}{49} \cdot \frac{39}{52} \cdot \frac{38}{51} \cdot \frac{37}{50} = \frac{13}{49} \cdot \frac{1}{4} \cdot \frac{38}{17} \cdot \frac{37}{50} = \\
= \frac{13}{49} \cdot \frac{1}{2} \cdot \frac{19}{17} \cdot \frac{37}{50}.
\end{array} P ( D ) = P ( B ∣ A , B , C ) ⋅ P ( A , B , C ) = 49 13 ⋅ 52 52 − 13 ⋅ 51 51 − 13 ⋅ 50 50 − 13 = 49 13 ⋅ 52 39 ⋅ 51 38 ⋅ 50 37 = 49 13 ⋅ 4 1 ⋅ 17 38 ⋅ 50 37 = = 49 13 ⋅ 2 1 ⋅ 17 19 ⋅ 50 37 .
The respective expectation for the fourth person is
E ( D ) = x ⋅ P ( D ) = 700 ⋅ 13 ⋅ 19 ⋅ 37 49 ⋅ 17 ⋅ 100 ≈ 76.80. E(D) = x \cdot P(D) = 700 \cdot \frac{13 \cdot 19 \cdot 37}{49 \cdot 17 \cdot 100} \approx 76.80. E ( D ) = x ⋅ P ( D ) = 700 ⋅ 49 ⋅ 17 ⋅ 100 13 ⋅ 19 ⋅ 37 ≈ 76.80.
Let F = F = F = F = { 700 , p 1 = 1 / 4 700 , p 2 = 13 / ( 17 ⋅ 4 ) 700 , p 3 = ( 13 ⋅ 19 ) / ( 17 ⋅ 100 ) 700 , p 4 = ( 13 ⋅ 19 ⋅ 37 ) / ( 49 ⋅ 17 ⋅ 100 ) F = \begin{cases} 700, & p_1 = 1/4 \\ 700, & p_2 = 13/(17 \cdot 4) \\ 700, & p_3 = (13 \cdot 19)/(17 \cdot 100) \\ 700, & p_4 = (13 \cdot 19 \cdot 37)/(49 \cdot 17 \cdot 100) \end{cases} F = ⎩ ⎨ ⎧ 700 , 700 , 700 , 700 , p 1 = 1/4 p 2 = 13/ ( 17 ⋅ 4 ) p 3 = ( 13 ⋅ 19 ) / ( 17 ⋅ 100 ) p 4 = ( 13 ⋅ 19 ⋅ 37 ) / ( 49 ⋅ 17 ⋅ 100 )
The respective expectation is
E ( F ) = 700 p 1 + 700 p 2 + 700 p 3 + 700 p 4 = 700 ⋅ ( 1 4 + 13 17 ⋅ 4 + 13 ⋅ 19 17 ⋅ 100 + 13 ⋅ 19 ⋅ 37 49 ⋅ 17 ⋅ 100 ) = 487.33. E(F) = 700p_1 + 700p_2 + 700p_3 + 700p_4 = 700 \cdot \left(\frac{1}{4} + \frac{13}{17 \cdot 4} + \frac{13 \cdot 19}{17 \cdot 100} + \frac{13 \cdot 19 \cdot 37}{49 \cdot 17 \cdot 100}\right) = 487.33. E ( F ) = 700 p 1 + 700 p 2 + 700 p 3 + 700 p 4 = 700 ⋅ ( 4 1 + 17 ⋅ 4 13 + 17 ⋅ 100 13 ⋅ 19 + 49 ⋅ 17 ⋅ 100 13 ⋅ 19 ⋅ 37 ) = 487.33.
Answer: 487.33.
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#340153 on Dec 2023
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