Question

The Hullian learning model asserts that the probability p of mastering a task after t learning trials is approximated by p (t) = 1 - e -kt where k is a constant that depends on the task to be learned. Suppose that a new dance is taught to an aerobics class. For this particular dance, the constant k = 0.28.

a. What is the probability of mastering the dance's steps in 1 trial? 2 trials? 5 trials? 11 trials? 16 trials? 20 trials?
Find the rate of change, p'(t).
Sketch a graph of the function.

Answer 1

Explanation:

a. Using the formula p(t) = 1 - e^(-kt), where k = 0.28:

Probability of mastering the dance's steps in 1 trial: p(1) = 1 - e^(-0.28*1) ≈ 0.243

Probability of mastering the dance's steps in 2 trials: p(2) = 1 - e^(-0.28*2) ≈ 0.446

Probability of mastering the dance's steps in 5 trials: p(5) = 1 - e^(-0.28*5) ≈ 0.846

Probability of mastering the dance's steps in 11 trials: p(11) = 1 - e^(-0.28*11) ≈ 0.981

Probability of mastering the dance's steps in 16 trials: p(16) = 1 - e^(-0.28*16) ≈ 0.997

Probability of mastering the dance's steps in 20 trials: p(20) = 1 - e^(-0.28*20) ≈ 0.999

b. The rate of change, p'(t), can be found by taking the derivative of the function p(t):

p'(t) = k * e^(-kt)

c. Here's a graph of the function p(t):

I apologize for the technical issue, but the graph doesnt load, so here's a description of the graph:

The graph of p(t) should be an increasing curve that starts at 0 and approaches 1 asymptotically. As t increases, the rate of change of p(t) decreases, which means that the curve becomes flatter and approaches the horizontal asymptote of y=1. The curve is concave down, meaning that its rate of change is decreasing. At t=0, the rate of change is k, which is the steepest point of the curve.

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graph of p(t) = 1 - e^(-0.28*t)

The x-axis represents the number of learning trials, and the y-axis represents the probability of mastering the dance's steps. As the number of trials increases, the probability of mastering the steps approaches 1 (or 100%).

Answer 2

The probability of mastering the dance after a specific number of trials is computed using the Hullian learning model with k=0.28. The probability increases with each additional trial, and the rate of change is the derivative of the probability function, showing how the rate of learning changes per trial.

The Hullian learning model is based on the function p(t) = 1 - e-kt where k is a constant specific to the task. In this scenario, we are given k = 0.28 for learning a new dance. To calculate the probability of mastering the dance after a certain number of trials, we substitute the number of trials into the function.

For example:

• After 1 trial: p(1) = 1 - e-(0.28)(1)
• After 2 trials: p(2) = 1 - e-(0.28)(2)
• After 5 trials: p(5) = 1 - e-(0.28)(5)

The rate of change of mastering the task, p'(t), is found by deriving the function. We get p'(t) = ke-kt, which for k = 0.28 becomes p'(t) = 0.28e-(0.28)t.

To sketch the graph, we plot p(t) against t and observe the exponential learning curve approaching the probability of 1 (indicating mastery of the dance) as trial number increases. The rate of change will decay as t increases, demonstrating a slowing in the rate of learning over time.