Binomial Distribution Finding P On Ti 84 Plus Calculator

This tool calculates binomial probabilities (like `binompdf` and `binomcdf` on your TI 84 Plus) and the article below explains how to approach finding ‘p’ (the probability of success) using your calculator.

Binomial Probability Calculator (TI 84 Plus Style)

k	P(X=k)	P(X≤k)
Enter values and calculate

Table of individual (P(X=k)) and cumulative (P(X≤k)) binomial probabilities.

What is Binomial Distribution Finding ‘p’ on TI 84 Plus Calculator?

The phrase “binomial distribution finding p on TI 84 Plus calculator” refers to the process of determining the probability of success (‘p’) in a series of independent trials, given a certain number of trials (‘n’), a number of successes (‘x’), and usually either the exact probability P(X=x) or the cumulative probability P(X≤x), using the functions available on a Texas Instruments TI-84 Plus graphing calculator.

The TI-84 Plus has built-in functions like `binompdf(n,p,x)` which calculates P(X=x) and `binomcdf(n,p,x)` which calculates P(X≤x). However, it doesn’t directly solve for ‘p’ if you know ‘n’, ‘x’, and the probability. “Finding p” usually involves using these functions in conjunction with the TI-84’s Solver or by iterating through values of ‘p’ until the desired probability is achieved.

This is useful for statisticians, students, and researchers who have observed a certain number of successes in a fixed number of trials and want to infer the underlying probability of success per trial.

Common Misconceptions:

• The TI-84 Plus has a direct “inverse binomial” function to find ‘p’. (It doesn’t directly solve for ‘p’ given probability, n, and x; you typically use the Solver).
• Finding ‘p’ is always a single-step calculation. (It often requires iterative methods or solving an equation).

Binomial Distribution Formulas and the TI-84 Plus

The core formulas used by the TI-84 Plus for binomial distributions are:

1. Binomial Probability Density Function (PDF) – `binompdf(n,p,x)`:

P(X = x) = _nC_x * p^x * (1-p)^n-x

This calculates the probability of getting exactly ‘x’ successes in ‘n’ trials.

2. Binomial Cumulative Distribution Function (CDF) – `binomcdf(n,p,x)`:

P(X ≤ x) = ∑_i=0^x [_nC_i * pⁱ * (1-p)^n-i]

This calculates the probability of getting at most ‘x’ successes (from 0 to x) in ‘n’ trials.

Where:

• _nC_x = n! / (x! * (n-x)!) is the number of combinations.
• ‘n’ is the number of trials.
• ‘p’ is the probability of success on one trial.
• ‘x’ is the number of successes.
• (1-p) is the probability of failure on one trial.

To find ‘p’ when you know ‘n’, ‘x’, and either P(X=x) or P(X≤x), you would set up an equation using one of the formulas above and solve for ‘p’. On the TI-84 Plus, this is typically done using the “Solver” feature (often found under the MATH menu).

For example, if you know P(X=x) = 0.1, n=10, x=2, you would set up: 0.1 = ₁₀C₂ * p² * (1-p)⁸ and solve for ‘p’ using the TI-84 Solver.

Variables Table

Variable	Meaning	Unit	Typical Range
n	Number of independent trials	Count	≥ 0 (integer)
p	Probability of success on a single trial	Probability	0 to 1
x or k	Number of successful outcomes	Count	0 to n (integer)
P(X=x)	Probability of exactly x successes	Probability	0 to 1
P(X≤x)	Cumulative probability of at most x successes	Probability	0 to 1

Practical Examples (Finding ‘p’ on TI 84 Plus)

Let’s illustrate how you would approach binomial distribution finding p on ti 84 plus calculator scenarios.

Example 1: Finding ‘p’ from P(X=x) using TI-84 Solver

Suppose you conduct 20 trials (n=20) and observe exactly 5 successes (x=5). You have reason to believe the probability of exactly 5 successes, P(X=5), was 0.176. What was the approximate probability of success ‘p’ on each trial?

1. On your TI-84 Plus, go to the Solver. Press `MATH`, then scroll up or down to `Solver…` (it might be `B:` or `0:` depending on the OS version).
2. You need to enter the equation: `0 = binompdf(20, P, 5) – 0.176`. Here, we use ‘P’ as the variable to solve for (our ‘p’). We rearrange P(X=5) = 0.176 to binompdf(n,p,x) – P(X=5) = 0.
   You enter it as `0 = binompdf(20,X,5)-.176` (using X for p in the solver). Access `binompdf` via `2nd` `VARS` [DISTR] menu.
3. Enter an initial guess for ‘P’ (or X on the screen), say 0.5.
4. Move the cursor to the line with ‘X=’ (or ‘P=’) and press `ALPHA` `ENTER` [SOLVE].
5. The calculator will find a value for ‘P’ that makes the equation true. It might be around 0.25. So, p ≈ 0.25.

This demonstrates the process of binomial distribution finding p on ti 84 plus calculator when P(X=x) is known.

Example 2: Finding ‘p’ from P(X≤x) using TI-84 Solver

Imagine 15 trials (n=15) were conducted. The cumulative probability of getting at most 3 successes, P(X≤3), was found to be 0.648. What is ‘p’?

1. Go to the TI-84 Solver.
2. Enter the equation: `0 = binomcdf(15, P, 3) – 0.648` (or `0 = binomcdf(15,X,3)-.648` using X for p). Access `binomcdf` via `2nd` `VARS` [DISTR].
3. Enter a guess for ‘P’ (or X), maybe 0.3.
4. Solve for ‘P’ (or X). You should get p ≈ 0.20.

This is another way of approaching binomial distribution finding p on ti 84 plus calculator, this time with cumulative probability.

How to Use This Calculator and Your TI 84 Plus

Using Our Calculator:

1. Enter n, p, and x: Input the number of trials (n), the probability of success (p), and the number of successes (x) into the fields above.
2. View Results: The calculator instantly shows P(X=x) (like `binompdf`) and P(X≤x) (like `binomcdf`), along with the distribution table and chart.
3. Understand TI-84 Functions: Our calculator mirrors what `binompdf(n,p,x)` and `binomcdf(n,p,x)` would show on your TI-84 for the given n, p, and x.

Finding ‘p’ on Your TI 84 Plus:

1. Identify Knowns: Determine ‘n’, ‘x’, and the known probability (either P(X=x) or P(X≤x)).
2. Access Solver: Press `MATH` and select `Solver…`.
3. Enter Equation:
   • If P(X=x) is known: `0 = binompdf(n, X, x) – P(X=x)` (replace n, x, P(X=x) with values, use X for ‘p’).
   • If P(X≤x) is known: `0 = binomcdf(n, X, x) – P(X≤x)` (replace n, x, P(X≤x) with values, use X for ‘p’).

   Find `binompdf` and `binomcdf` under `2nd` `VARS` [DISTR].

4. Initial Guess: Enter an estimated value for X (which represents ‘p’, so between 0 and 1).
5. Solve: Place the cursor on the X= line and press `ALPHA` `ENTER` [SOLVE]. The calculator will find ‘p’.

This iterative approach using the Solver is key to binomial distribution finding p on ti 84 plus calculator tasks.

Key Factors That Affect Binomial Probabilities and Finding ‘p’

• Number of Trials (n): A larger ‘n’ generally makes the distribution more spread out, and the probability of any single ‘x’ value lower, unless ‘p’ is very close to 0 or 1. Finding ‘p’ can be more sensitive with larger ‘n’.
• Probability of Success (p): This is what you are often trying to find. The closer ‘p’ is to 0.5, the more symmetric the distribution. Values of ‘p’ near 0 or 1 create skewed distributions.
• Number of Successes (x): The target number of successes relative to ‘n’ and ‘p’ determines the probability P(X=x) or P(X≤x).
• Known Probability Value: The accuracy of the given P(X=x) or P(X≤x) directly impacts the accuracy of the ‘p’ you find.
• Solver’s Initial Guess (TI-84): While the TI-84 Solver is robust, a very poor initial guess for ‘p’ might lead to a longer solution time or, rarely, a non-convergence or finding an unintended solution if the equation is complex.
• Calculator Precision: The TI-84 Plus works with a high degree of precision, but rounding in intermediate steps (if done manually) could affect the final ‘p’. Using the `binompdf/cdf` functions directly in the Solver is best.

Frequently Asked Questions (FAQ)

Q1: Does the TI-84 Plus have a direct function to find ‘p’ given n, x, and probability?

A1: No, the TI-84 Plus does not have a direct “inverse binomial” function to solve for ‘p’. You typically use the `Solver` with `binompdf` or `binomcdf` to find ‘p’ when ‘n’, ‘x’, and a probability are known.

Q2: What is `binompdf` on the TI-84 Plus?

A2: `binompdf(n,p,x)` calculates the probability of getting *exactly* ‘x’ successes in ‘n’ trials, given a probability of success ‘p’.

Q3: What is `binomcdf` on the TI-84 Plus?

A3: `binomcdf(n,p,x)` calculates the cumulative probability of getting *at most* ‘x’ successes (from 0 to x) in ‘n’ trials, given ‘p’.

Q4: How do I access `binompdf` and `binomcdf` on my TI-84 Plus?

A4: Press `2nd` then `VARS` (to get to the [DISTR] menu), then scroll down to find `binompdf(` and `binomcdf(`.

Q5: How do I access the Solver on the TI-84 Plus?

A5: Press the `MATH` button, then scroll up or down until you find `Solver…` and press `ENTER`.

Q6: What if the Solver takes too long or gives an error when finding ‘p’?

A6: Try a different initial guess for ‘p’ (between 0 and 1). Ensure your equation is entered correctly. For `binompdf` and `binomcdf`, ‘n’ must be a positive integer, ‘x’ an integer between 0 and n, and ‘p’ between 0 and 1.

Q7: Can I find ‘p’ if I know P(X ≥ x)?

A7: Yes. P(X ≥ x) = 1 – P(X ≤ x-1). So, if you know P(X ≥ x), you know P(X ≤ x-1). You can then use `binomcdf(n, p, x-1)` in the Solver. For example, if P(X ≥ 5) = 0.3, then P(X ≤ 4) = 0.7. Use `0 = binomcdf(n, X, 4) – 0.7` in the solver.

Q8: Is this calculator the same as using the TI-84 Plus for finding p?

A8: Our calculator shows you the results of `binompdf` and `binomcdf` for given n, p, and x. The article explains the process of using these, along with the TI-84’s Solver, to actually find ‘p’ when it’s unknown. Our calculator doesn’t solve for ‘p’ directly but helps understand the functions you’d use on the TI-84.