Binomial Problem: I Don't Understand i=6

[nacho-man]

please refer to attached image

I don't understand the reasoning for i=6

since a qualified candidate must have answered 15+ questions correctly, we have
i being summed from 15 to 20 for the first part.

shouldn't the second part be summed from i=15 to 20?

for example, $(20,6)$ would imply, 6 successes, and 14 failures in a question. This would make a candidate unsuccessful by the quiz and, would give us $P(A^c|Q^c)$ instead of $P(A|Q^c)$ ?

 

[mmwave]

Hello,

Thank you for your question. I understand your confusion about the reasoning for i=6 in the formula shown in the attached image. Let me provide some clarification.

Firstly, the formula shown in the image is the formula for conditional probability, specifically $P(A|Q^c)$, which represents the probability of event A occurring given that event Q does not occur. In this context, event A represents a candidate being qualified for a quiz and event Q represents a candidate answering less than 15 questions correctly.

Now, let's break down the formula. The summation symbol $\sum$ indicates that we are summing up the probabilities of all possible outcomes for the given scenario. In this case, we are summing up the probabilities of all possible combinations of successes and failures, represented by i and j respectively.

The first part of the formula, $\sum_{i=15}^{20} \binom{20}{i}(0.8)^i(0.2)^{20-i}$, represents the sum of probabilities for all combinations where the candidate answers 15 or more questions correctly, which is the minimum requirement for a candidate to be qualified. This is why the summation starts at i=15, as we are only considering combinations where i (number of successes) is equal to or greater than 15.

Now, the second part of the formula, $\sum_{i=6}^{14} \binom{20}{i}(0.8)^i(0.2)^{20-i}$, represents the sum of probabilities for all combinations where the candidate answers less than 15 questions correctly. This is where the i=6 comes in. Since we are only considering combinations where the candidate has less than 15 successes, the value of i starts from 6, as this is the minimum number of successes needed for the candidate to be considered unsuccessful.

In summary, the formula is correct as it is written and the second part should indeed be summed from i=6 to 14. I hope this helps to clarify the reasoning behind the formula. Let me know if you have any further questions or concerns. Thank you.