We need to find the probability that James answers more than 3 but less than 10 questions correctly out of 15, given that each question has a 1/3 chance of being answered correctly. This is a binomial distribution problem.

Since the inequalities are not inclusive and X is discrete:

3<X<10\Longrightarrow4\le X\le9

In other words, we are looking for

\begin{align}\mathop{\mathrm{P}}(X=4,5,6,7,8,9) & =\mathop{\mathrm{P}}(X=1,2,3,4,5,6,7,8,9)-\mathop{\mathrm{P}}(X=1,2,3)\\ \placeholder{} & \placeholder{}\\ \placeholder{} & =\mathop{\mathrm{P}}(X\le9)-\mathop{\mathrm{P}}(X\le3)\end{align}

The probability of X being less than or equal to a certain value in a binomial distribution is calculated using the binomial cumulative density function (CDF):

\mathop{\mathrm{P}}(X=4,5,6,7,8,9)=\mathrm{binomcdf}\left(15,\frac13,9\right)-\mathrm{binomcdf}\left(15,\frac13,3\right)

Here, 15 is the number of trials and \frac13 is the probability of success.

Here are the steps to calculate this probability using a TI-84 calculator:

Turn on the Calculator: Press the ON button.
Access the Distribution Menu: Press the 2nd button, then VARS to access the DISTR menu.
Select the Binomial CDF Function: Scroll down to find binomcdf(. This function is used to find the cumulative probability of a certain number of successes or fewer in a binomial distribution. Press ENTER to select it.
Calculate the Probability for X \leq 3:
- Enter the parameters for the cumulative distribution function: the number of trials (n), the probability of success (p), and the number of successes (x) you are interested in. For X \leq 3:
  - Number of trials, n = 15
  - Probability of success, p = \frac{1}{3}
  - Number of successes, x = 3
- Enter these values in the format binomcdf(15, 1/3, 3) =0.209
- Press ENTER to calculate the cumulative probability for X \leq 3.
Calculate the Probability for X \leq 9 :
- Repeat the process for X \leq 9:
  - Enter binomcdf(15, 1/3, 9).
- Press ENTER to calculate the cumulative probability for X \leq 9.
Find the Probability for 3 < X < 10:
- Subtract the probability of X \leq 3 from the probability of X \leq 9 to find the probability that 3 < X < 10.
- This is done by calculating: binomcdf(15, 1/3, 9) - binomcdf(15, 1/3, 3) =0.782

The result will give you the probability that James answers more than 3 but less than 10 questions correctly.

r	1	2	3	4	5	6
P(R=r)	0.1	0.1	p	q	0.2	0.2

r	1	2	3	4	5	6
P(R=r)	0.1	0.1	p	q	0.2	0.2

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