Question

Based on the given definitions of BA, MBA$_1$, and MBA$_2$, which of the following is true?

• A. MBA$_1 \le$ BA $\le$ MBA$_2$
• B. BA $\le$ MBA$_2 \le$ MBA$_1$
• C. MBA$_2 \le$ BA $\le$ MBA$_1$
• D. None of these
• A. BA and MBA$_1$ will both increase
• B. BA will increase and MBA$_1$ will decrease
• C. BA will increase and not enough data is available to assess change in MBA$_1$ and MBA$_2$
• D. None of these

Answer 1

The Correct Option is B

Given: $\text{BA} = \frac{r_1 + r_2}{n_1}$, $\text{MBA}_1 = \frac{r_1}{n_1} + \frac{n_2}{n_1} \max\left[0, \frac{r_2}{n_2} - \frac{r_1}{n_1}\right]$, $\text{MBA}_2 = \frac{r_1 + r_2}{n_1 + n_2}$. From structure: - BA uses all runs divided by completed innings only, hence BA $\ge$ MBA$_2$ (since MBA$_2$ divides by total innings, which is larger). - MBA$_1$ is constructed to be at least as large as BA, since the adjustment term is non-negative. Therefore BA $\le$ MBA$_2 \le$ MBA$_1$ holds.

Answer 2

Initially no incomplete innings: BA = $\frac{r_1}{n_1} = 50$. Adding an incomplete innings with $r_2 = 45$, BA formula $\frac{r_1 + r_2}{n_1}$ increases numerator without changing $n_1$, so BA increases. However, MBA$_1$ depends on comparison between $\frac{r_2}{n_2}$ and $\frac{r_1}{n_1}$, which needs $n_2$ and prior $r_2$ values. MBA$_2$ also depends on $n_2$ and may increase or decrease. Hence, we cannot conclude changes in MBA$_1$ or MBA$_2$.