Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is :
Solution
Out of 11 consecutive natural numbers either
6 even and 5 odd numbers or 5 even and 6
odd numbers.
<br><br>Let, E = Even
<br>O = Odd
<br><br><b>Case-1 :</b>
<br><br>E, O, E, O, E, O, E, O, E, O, E
<br><br>2b = a + c $\Rightarrow$ Even
<br><br>$\Rightarrow$ Both a and c should be either even or odd.
<br><br>P = ${{{}^6{C_2} + {}^5{C_2}} \over {{}^{11}{C_3}}}$ = ${5 \over {33}}$
<br><br><b>Case -2 :</b>
<br><br>O, E, O, E, O, E, O, E, O, E, O
<br><br>P = ${{{}^5{C_2} + {}^6{C_2}} \over {{}^{11}{C_3}}}$ = ${5 \over {33}}$
<br><br>Total probability = ${1 \over 2} \times {5 \over {33}}$ + ${1 \over 2} \times {5 \over {33}}$ = ${5 \over {33}}$
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
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