A final year student appears for placement interview in two companies, S and T. Based on her interview performance, she estimates the probability of receiving job offers from companies S and T to be 0.8 and 0.6, respectively. Let p be the probability that she receives job offers from both the companies. Select the most appropriate option.

The correct option is 0.4 ≤ p ≤ 0.6.

Let us analyze the problem step-by-step to understand why this range for the probability $p$ is correct.

Let $S$ be the event that the student receives a job offer from company S, and $T$ be the event that she receives a job offer from company T.
We are given the individual probabilities of these events:
$P\left(S\right)=0.8$
$P\left(T\right)=0.6$
Let $p$ be the probability that she receives job offers from both companies. This is represented by the intersection of the two events:
$p=P(S\cap T)$

To find the bounds for $p$, we use standard probability inequalities.

1. Finding the Upper Bound:
The probability of the intersection of two events cannot exceed the probability of either individual event.
Therefore, we must have:
$P(S\cap T)\le P\left(S\right)$ and $P(S\cap T)\le P\left(T\right)$
Thus, $p\le min\left(P\right(S),P(T\left)\right)$
Substituting the given values:
$p\le min(0.8,0.6)=0.6$

2. Finding the Lower Bound:
We know that the probability of the union of two events is given by the formula:
$P(S\cup T)=P\left(S\right)+P\left(T\right)-P(S\cap T)$
Rearranging this equation to solve for $P(S\cap T)$, we get:
$P(S\cap T)=P\left(S\right)+P\left(T\right)-P(S\cup T)$
Since the probability of any union of events cannot exceed 1:
$P(S\cup T)\le 1$
Using this inequality, we can write:
$P(S\cap T)\ge P\left(S\right)+P\left(T\right)-1$
Substituting the given values:
$p\ge 0.8+0.6-1$
$p\ge 1.4-1=0.4$

Conclusion:
Combining the lower bound and the upper bound, we get:
$0.4\le p\le 0.6$