We are given:
Probability of passing = p
Probability of Failing = 6p²
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Finding the value of 'p':
We know that the sum of the probabilities of an event occurring and not occurring is 1
So, Probability of passing + Probability of Failing = 1
p + 6p² = 1
6p² + p - 1 = 0
We can see that we now have a quadratic equation
Splitting the middle term:
6p² + 3p - 2p - 1 = 0
3p(2p + 1) - (2p + 1) = 0
(3p - 1)(2p + 1) = 0 [factoring out 2p + 1]
Now, we can either divide both sides by (3p-1) or by (2p + 1) and both the cases will give us different answers.
Dividing both sides by (3p - 1):
2p + 1 = 0/(3p-1)
2p + 1 = 0
p = -1/2
Dividing both sides by (2p+1):
(3p - 1) = 0 / (2p + 1)
3p - 1 = 0
p = 1/3
Now, we have 2 values of p: -1/2 and 1/3
Using the value: -1/2 will mean that the probabilities are negative
but we know that probabilities cannot be negative.
So, -1/2 is not a valid value for p
Therefore, the value of p is 1/3
