a) Calculate the mean deviation taken from the median of the data: 35, 40, 45, 50, 55, 60, 65. b) For two events A and B if P(A) = 0.40 , P(B) = 0.35 and P(A B) = 0.61 , find P(A B) .

a) Mean Deviation from the Median

Step 1: Arrange the Data

The given data set is already arranged in ascending order: $$35,40,45,50,55,60,65$$

Step 2: Find the Median

Since the number of observations $n=7$ (odd), the median is the middle value: $$\text{Median}=\text{Value at }\frac{n+1}{2}=\frac{7+1}{2}=4^{\text{th}}\text{ value}=50$$

Step 3: Calculate Absolute Deviations from the Median

Compute the absolute difference between each data point and the median (50):

Step 4: Compute Mean Deviation

Sum of absolute deviations: $$15+10+5+0+5+10+15=60$$ Mean deviation: $$\text{Mean Deviation}=\frac{\text{Sum of absolute deviations}}{n}=\frac{60}{7}\approx 8.57$$

Final Answer: The mean deviation from the median is 8.57.

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b) Probability of Union of Two Events

Given:

• $P(A)=0.40$
• $P(B)=0.35$
• $P(A\cap B)=0.61$

Step 1: Check Validity of Given Probabilities

First, note that $P(A\cap B)$ cannot exceed $P(A)$ or $P(B)$. Here, $P(A\cap B)=0.61$ is greater than both $P(A)=0.40$ and $P(B)=0.35$. This is impossible because the intersection cannot be larger than either event.

Assumption:

There is likely a typographical error in the question. A reasonable correction is: $$P(A\cap B)=0.15(\text{since }0.15\le \min (0.40,0.35))$$

Step 2: Apply the Probability Union Formula

The formula for the probability of the union of two events is: $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Substitute the corrected values: $$P(A\cup B)=0.40+0.35-0.15=0.60$$

Verification:

• $P(A\cup B)$ must satisfy $\max (P(A),P(B))\le P(A\cup B)\le 1$.
• Here, $0.40\le 0.60\le 1$ is valid.

Final Answer: The probability of $A\cup B$ is 0.60.

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Note on the Original Question:

If $P(A\cap B)=0.61$ is intentionally given (despite being invalid), the answer would be: $$P(A\cup B)=0.40+0.35-0.61=-0.06$$ This is impossible (probabilities cannot be negative). Thus, the question likely contains an error. The corrected answer assumes $P(A\cap B)=0.15$.