| Direction: Assertion- Reaction type. Each of these contains tow statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes [a], [b], [c] and [d] given below: |
| Statement I: A coin is tossed 31 times. If the probability of getting number of heads more than the number of tails is equal to the probability of getting tails more than the number of heads, then the coin must be unbiased. |
| Statement II: If p = q and p + q = 1, then coin is unbiased. |
A) Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
B) Statement I is true; Statement II is false.
C) Statement I is false; Statement II is true.
D) Statement I is true; Statement II is true; Statements is the correct explanation for Statement I.
Correct Answer: D
Solution :
Given that, n =31 and \[p(X\ge 16)=p(X\le 15)\] \[\Rightarrow \]\[^{31}{{C}_{16}}{{p}^{16}}{{q}^{15}}{{+}^{31}}{{C}_{17}}{{p}^{17}}{{q}^{14}}+\]?.\[{{+}^{31}}{{C}_{31}}{{p}^{31}}\] \[{{=}^{31}}{{C}_{0}}{{q}^{31}}{{+}^{31}}{{C}_{1}}p{{q}^{30}}{{+}^{31}}{{C}_{2}}{{p}^{2}}{{q}^{29}}+\]?..\[{{+}^{31}}{{C}_{15}}{{p}^{15}}{{q}^{16}}\] \[\Rightarrow \]\[^{31}{{C}_{31}}{{p}^{31}}+...{{+}^{31}}{{C}_{17}}{{p}^{17}}{{q}^{14}}{{+}^{31}}{{C}_{16}}{{p}^{16}}{{q}^{15}}\] \[{{=}^{31}}{{C}_{0}}{{q}^{31}}{{+}^{31}}{{C}_{1}}p{{q}^{30}}{{+}^{31}}{{C}_{2}}{{p}^{2}}{{q}^{29}}+...\]\[{{+}^{31}}{{C}_{15}}{{p}^{15}}{{q}^{16}}\] \[\Rightarrow \]\[^{31}{{C}_{0}}{{p}^{31}}{{+}^{31}}{{C}_{1}}{{p}^{30}}q+{{...}^{31}}{{C}_{14}}{{p}^{17}}{{q}^{14}}\]\[{{+}^{31}}{{C}_{15}}{{p}^{16}}{{q}^{15}}\] \[{{=}^{31}}{{C}_{0}}{{q}^{31}}{{+}^{31}}{{C}_{1}}p{{q}^{30}}{{+}^{31}}{{C}_{2}}{{p}^{2}}{{q}^{29}}+...\] \[{{+}^{31}}{{C}_{15}}{{p}^{15}}{{q}^{16}}\] \[\Rightarrow \]\[^{31}{{C}_{0}}({{p}^{31}}={{q}^{31}}){{+}^{31}}{{C}_{1}}pq({{p}^{29}}-{{q}^{29}})+\]?..\[{{+}^{31}}{{C}_{15}}{{p}^{15}}{{q}^{15}}(p-q)=0\] \[\Rightarrow \]\[(p-q)\lambda =0\] Where,\[\lambda {{=}^{31}}{{C}_{0}}({{p}^{30}}+{{p}^{29}}q+{{p}^{28}}{{q}^{2}}+....+{{q}^{30}})\] \[{{+}^{31}}{{C}_{1}}pq({{p}^{28}}+{{p}^{27}}q+....+{{q}^{28}})+...\] \[{{+}^{31}}{{C}_{15}}{{p}^{15}}{{q}^{15}}\] As\[\lambda >0\therefore p-q=0\Rightarrow p=q\] \[\Rightarrow \]\[p=q=\frac{1}{2}\] \[(\therefore p+q=1)\] Hence, the coin is unbiased.
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