If P(X ≤ 4)=0.3 for a normal random variable, what is P(X > 2μ−4)?

Expert intro: For a normal random variable, points reflected about the mean have mirrored cumulative probabilities. That symmetry does most of the work here.

Detailed walkthrough

Let

$X\sim N(\mu,\sigma^2).$

We are told that

$P(X\le 4)=0.3.$

Step 1: Convert 4 into a z-value idea

Since the normal distribution is symmetric about its mean, the point reflected across the mean from 4 is

$2\mu-4.$

If 4 is at the 30th percentile, then its reflection is at the 70th percentile.

So,

$P(X\le 2\mu-4)=0.7.$

Step 2: Take the upper tail

Therefore,

$P(X>2\mu-4)=1-0.7=0.3.$

So the required probability is

$\boxed{0.3}.$

💡 Pitfall guide

A common mistake is to stop at the reflected CDF value and forget the question asks for the probability greater than $2\mu-4$, not less than or equal to it. Another trap is mixing up the symmetry point and thinking the answer must be 0.7 because of the reflected percentile.

🔄 Real-world variant

If the question instead asked for $P(X<2\mu-4)$, the answer would be $0.7$. If the given probability were $P(X\le 4)=0.8$, then the reflected point would correspond to $0.2$ on the left tail and $0.8$ on the upper tail in the opposite direction.

🔍 Related terms

normal distribution, symmetry, percentiles