Question

A sample space consists of 3 sample points with associated probabilities given as $2p$, ${p^2}$, $4p - 1$ then
A) $p = \sqrt {11} - 3$
B) $p = \sqrt {10} - 3$
C) $\dfrac{1}{4} < p < \dfrac{1}{2}$
D) None

Answer 1

Here we need to find the range of the variable used in the question. We will first find the sum of all the probabilities of three events and then we will equate it with one. From there, we will get the equation including the variable and after simplifying the equation, we will get the value of the variable.

Formula used:
The roots of the quadratic equation say $a{x^2} + bx + c = 0$ is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.

Complete step by step solution:
Here we have been given the probabilities of events in a sample space i.e. $2p$, ${p^2}$ and $4p - 1$
We know that the probability of all the events is equal to 1.
So, we will add all the given probabilities and then equate it with one.
$2p + {p^2} + 4p - 1 = 1$
Adding the like terms, we get
$\Rightarrow {p^2} + 6p - 1 = 1$
Now, we will subtract the number 1 form both sides. Therefore, we get
$\Rightarrow {p^2} + 6p - 1 - 1 = 1 - 1$
On further simplification, we get
$\Rightarrow {p^2} + 6p - 2 = 0$
We can see that this is a quadratic equation.
We know that the formula to find the roots of the quadratic equation say $a{x^2} + bx + c = 0$ is given by
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now, substituting $a = 1,b = - 6$ and $c = - 2$ in the above formula, we get
$p = \dfrac{{ - 6 \pm \sqrt {{6^2} - 4 \times 1 \times - 2} }}{{2 \times 1}}$
Now, we will simplify the terms under the square root. Therefore, we get
$\Rightarrow p = \dfrac{{ - 6 \pm \sqrt {36 + 8} }}{2}$
Adding the terms, we get
$\Rightarrow p = \dfrac{{ - 6 \pm \sqrt {44} }}{2}$
Now, we will further simplify the square root term.
$\Rightarrow p = \dfrac{{ - 6 \pm 2\sqrt {11} }}{2}$
On dividing the number 2, we get
$\Rightarrow p = - 3 \pm \sqrt {11}$
We know that the value of probability is always greater than 0.
So we can say that
$4p - 1 > 0$
Now, we will add the number 1 to both sides of the inequality.
\Rightarrow 4p - 1 + 1 > 0 + 1 \\\ \Rightarrow 4p > 1 \\\
Now, we will divide both sides by 4. Therefore, we get
\Rightarrow \dfrac{{4p}}{4} > \dfrac{1}{4} \\\ \Rightarrow p > \dfrac{1}{4} \\\
Therefore, we can say that the value of $p$ will be equal to $\sqrt {11} - 3$ but not equal to $- \sqrt {11} - 3$ as $- \sqrt {11} - 3$ is less than $\dfrac{1}{4}$.
Hence, the required value of $p$ is equal to $\sqrt {11} - 3$.

Therefore, the correct option is option A.

Note:
Probability is defined as the ratio of number of desired outcomes to the total number of possible outcomes. We need to keep in mind that the value of probability cannot be greater than 1 and also cannot be less than 0 i.e. negative. The probability of a sure event is always one, whereas if the probability of an event is 0, we can say that it is an impossible event. In real life, we use probability for weather forecasting, predicting scores in a cricket match etc.