Math Problem Statement

Medical research has shown tgat a certain type of chemotherapy is successful 70% of the time when used to treat skin cancer.suppose 5 skin cancer patients are treated eith thid type of chemotherapy and let x equal the no of successful cureaWhen x is 0 p(x) is 0.002, when x is 1 p(x) is 0.029 when x is 2 p(x) is 0.132 when x is 3 p(x) is 0.309 when its 4 p(x) is 0.360 when its 5 p(x) is 0 168.based on the topic discrete random variables a)verify that this is a probability distribution giving 3 reasons b) plot the probability histogram of the distribution c) what is the most likely no of patients successfully cured out of 5 d)what is the probability of getting 4 cured patients out of 5 ? Is this likely to occur e) what is the probability of getting atleast 4 cured patients f) what is the probability of getting less than 4cured patients g)what is the probability of getting 3 or less cured patients?

Solution

Here is the probability histogram showing the distribution of successful cures for 5 patients treated with chemotherapy. The x-axis represents the number of successful cures, and the y-axis represents the probability for each outcome.

(c) Most Likely Number of Patients Successfully Cured

The most likely number of patients successfully cured corresponds to the value of xx with the highest probability. From the dataHere is the probability histogram showing the distribution of successful cures for 5 patients treated with chemotherapy. The x-axis represents the number of successful cures, and the y-axis represents the probability for each outcome.

(c) Most Likely Number of Patients Successfully Cured

The most likely number of patients successfully cured corresponds to the value of xx with the highest probability. From the

  • P(4)=0.360P(4) = 0.360 is the highest value.

Thus, the most likely number of patients successfully cured is 4.

(d) Probability of Getting 4 Cured Patients out of 5

The probability of getting exactly 4 cured patients out of 5 is P(4)=0.360P(4) = 0.360. Since this is the highest probability in the distribution, it is the most likely outcome, meaning it is quite likely to occur.

(e) Probability of Getting At Least 4 Cured Patients

To find the probability of getting at least 4 cured patients, we sum the probabilities of getting 4 or 5 patients cured: P(X≥4)=P(4)+P(5)=0.360+0.168=0.528P(X \geq 4) = P(4) + P(5) = 0.360 + 0.168 = 0.528 The probability of getting at least 4 cured patients is 0.528.

(f) Probability of Getting Less Than 4 Cured Patients

To find the probability of getting less than 4 cured patients, we sum the probabilities of getting 0, 1, 2, or 3 patients cured: P(X<4)=P(0)+P(1)+P(2)+P(3)=0.002+0.029+0.132+0.309=0.472P(X < 4) = P(0) + P(1) + P(2) + P(3) = 0.002 + 0.029 + 0.132 + 0.309 = 0.472 The probability of getting less than 4 cured patients is 0.472.

(g) Probability of Getting 3 or Fewer Cured Patients

This is the probability of getting 0, 1, 2, or 3 cured patients: P(X≤3)=P(0)+P(1)+P(2)+P(3)=0.002+0.029+0.132+0.309=0.472P(X \leq 3) = P(0) + P(1) + P(2) + P(3) = 0.002 + 0.029 + 0.132 + 0.309 = 0.472 Thus, the probability of getting 3 or fewer cured patients is 0.472.

Would you like any more details or clarifications?

Here are 5 questions to expand your understanding:

  1. How is a probability distribution different from a frequency distribution?
  2. What does it mean if a random variable is "discrete"?
  3. How would the probability distribution change if the success rate of the chemotherapy improved to 80%?
  4. What other types of charts can be used to visualize probability distributions?
  5. How would you calculate the mean (expected value) of this distribution?

Tip: The sum of probabilities for any valid distribution must always equal 1.

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Math Problem Analysis

Mathematical Concepts

Probability Distribution
Discrete Random Variables
Binomial Probability

Formulas

Sum of probabilities: ∑P(x) = 1
P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))
P(X ≥ k) = P(k) + P(k+1) + ...
P(X ≤ k) = P(0) + P(1) + ... + P(k)

Theorems

Probability Distribution Verification
Binomial Theorem

Suitable Grade Level

Grades 10-12

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