supposed that k is a random variable. given that P(-3.7<=k<= 3.7) = 0.525 and that P(k<3.75) = P(k>3.7), find P(k>3.7).
To find P(k > 3.7), we need to use the information given in the question. Let's break it down step by step:
1. Given that P(-3.7 ≤ k ≤ 3.7) = 0.525, we know that this probability represents the area under the probability density function (PDF) between -3.7 and 3.7.
2. Since the probability of the entire sample space is equal to 1, we can find P(k > 3.7) by subtracting P(-3.7 ≤ k ≤ 3.7) from 1.
3. Therefore, P(k > 3.7) = 1 - P(-3.7 ≤ k ≤ 3.7).
4. Before proceeding, let's rewrite P(-3.7 ≤ k ≤ 3.7) as P(k ≤ 3.7) - P(k < -3.7). This is because we want to find the probability between these two points.
5. Given that P(k < 3.75) = P(k > 3.7), we can use this information to rewrite the expression further. We have:
P(-3.7 ≤ k ≤ 3.7) = P(k ≤ 3.7) - P(k < -3.7)
But P(k < 3.75) = P(k > 3.7). So, P(k ≤ 3.7) - P(k < -3.7) = P(k < 3.75) - P(k < -3.7).
6. Now, we can rewrite P(k > 3.7) using this information from step 5. It becomes:
P(k > 3.7) = 1 - (P(k < 3.75) - P(k < -3.7)).
7. Since P(k < 3.75) = P(k > 3.7), we can simplify the expression in step 6 further:
P(k > 3.7) = 1 - (P(k > 3.7) - P(k < -3.7)).
8. We have P(k > 3.7) on both sides of the equation. To solve for P(k > 3.7), we need to isolate it. Let's rearrange the equation:
P(k > 3.7) + P(k > 3.7) = 1 - P(k < -3.7)
2P(k > 3.7) = 1 - P(k < -3.7)
P(k > 3.7) = (1 - P(k < -3.7)) / 2.
9. Now, we can substitute the given probability from step 1 into the equation:
P(k > 3.7) = (1 - P(k < -3.7)) / 2 = (1 - 0.525) / 2 = 0.475 / 2 = 0.2375.
Therefore, P(k > 3.7) is equal to 0.2375.