Math / Data & Statistics

QuestionA new drug has shown to be effective in $68 \%$ of participants during the trials. In a group of 50 patients; what is the probability that the drug is not effective for at most 5 patients?
State answer as a decimal rounded to six decimal places. $\square$

Studdy Solution

STEP 1

What is this asking? Out of 50 patients, what's the chance that 0, 1, 2, 3, 4, or 5 patients find the drug ineffective? Watch out! Don't mix up "effective" and "not effective"!
Also, "at most 5" includes 5!

STEP 2

1. Find the probability of "not effective".
2. Set up the binomial probability formula.
3. Calculate the probabilities for 0 to 5 patients.
4. Sum the probabilities.

STEP 3

If the drug is effective for $68\%$ of patients, that means it's not effective for $100\% - 68\% = 32\%$ of patients.
So the probability of the drug being ineffective for a single patient is $0.32$ .
Let's call this probability $q = \textbf{0.32}$ .

STEP 4

We're dealing with a binomial probability here because each patient either finds the drug effective or not effective (success or failure).
The formula for binomial probability is: $P(X=k) = \binom{n}{k} \cdot p^k \cdot q^{n-k}$ Where: $n$ is the total number of trials (patients in our case, so $n = \textbf{50}$ ). $k$ is the number of "successes" (number of patients for whom the drug is not effective). $p$ is the probability of "success" on a single trial (probability of the drug being effective, which is $0.68$ ). $q$ is the probability of "failure" on a single trial (probability of the drug not being effective, which we found to be $q = \textbf{0.32}$ ). $\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$ .
It represents the number of ways to choose $k$ successes out of $n$ trials.

STEP 5

We need to calculate $P(X=k)$ for $k$ from 0 to 5.
Let's do it!

STEP 6

$P(X=0) = \binom{50}{0} \cdot (0.68)^0 \cdot (0.32)^{50-0} \approx \textbf{2.582249878086908e-20}$

STEP 7

$P(X=1) = \binom{50}{1} \cdot (0.68)^1 \cdot (0.32)^{50-1} \approx \textbf{9.399056546095877e-19}$

STEP 8

$P(X=2) = \binom{50}{2} \cdot (0.68)^2 \cdot (0.32)^{50-2} \approx \textbf{1.703826764584705e-17}$

STEP 9

$P(X=3) = \binom{50}{3} \cdot (0.68)^3 \cdot (0.32)^{50-3} \approx \textbf{2.058053823183734e-16}$

STEP 10

$P(X=4) = \binom{50}{4} \cdot (0.68)^4 \cdot (0.32)^{50-4} \approx \textbf{1.862613459045688e-15}$

STEP 11

$P(X=5) = \binom{50}{5} \cdot (0.68)^5 \cdot (0.32)^{50-5} \approx \textbf{1.343633738227269e-14}$

STEP 12

Now, we add up the probabilities we just calculated to find the probability that at most 5 patients find the drug ineffective: $P(X \le 5) = \sum_{k=0}^{5} P(X=k) \approx \textbf{1.57864e-14}$

STEP 13

The probability that the drug is not effective for at most 5 patients is approximately 0.0000000000000158.

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QuestionA new drug has shown to be effective in $68 \%$ of participants during the trials. In a group of 50 patients; what is the probability that the drug is not effective for at most 5 patients?
State answer as a decimal rounded to six decimal places. $\square$

Studdy Solution

STEP 1

What is this asking? Out of 50 patients, what's the chance that 0, 1, 2, 3, 4, or 5 patients find the drug ineffective? Watch out! Don't mix up "effective" and "not effective"!
Also, "at most 5" includes 5!

STEP 2

1. Find the probability of "not effective".
2. Set up the binomial probability formula.
3. Calculate the probabilities for 0 to 5 patients.
4. Sum the probabilities.

STEP 3

If the drug is effective for $68\%$ of patients, that means it's not effective for $100\% - 68\% = 32\%$ of patients.
So the probability of the drug being ineffective for a single patient is $0.32$ .
Let's call this probability $q = \textbf{0.32}$ .

STEP 4

STEP 5

We need to calculate $P(X=k)$ for $k$ from 0 to 5.
Let's do it!

STEP 6

$P(X=0) = \binom{50}{0} \cdot (0.68)^0 \cdot (0.32)^{50-0} \approx \textbf{2.582249878086908e-20}$

STEP 7

$P(X=1) = \binom{50}{1} \cdot (0.68)^1 \cdot (0.32)^{50-1} \approx \textbf{9.399056546095877e-19}$

STEP 8

$P(X=2) = \binom{50}{2} \cdot (0.68)^2 \cdot (0.32)^{50-2} \approx \textbf{1.703826764584705e-17}$

STEP 9

$P(X=3) = \binom{50}{3} \cdot (0.68)^3 \cdot (0.32)^{50-3} \approx \textbf{2.058053823183734e-16}$

STEP 10

$P(X=4) = \binom{50}{4} \cdot (0.68)^4 \cdot (0.32)^{50-4} \approx \textbf{1.862613459045688e-15}$

STEP 11

$P(X=5) = \binom{50}{5} \cdot (0.68)^5 \cdot (0.32)^{50-5} \approx \textbf{1.343633738227269e-14}$

STEP 12

Now, we add up the probabilities we just calculated to find the probability that at most 5 patients find the drug ineffective: $P(X \le 5) = \sum_{k=0}^{5} P(X=k) \approx \textbf{1.57864e-14}$

STEP 13

The probability that the drug is not effective for at most 5 patients is approximately 0.0000000000000158.

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QuestionA new drug has shown to be effective in 68%68 \%68% of participants during the trials. In a group of 50 patients; what is the probability that the drug is not effective for at most 5 patients?State answer as a decimal rounded to six decimal places. □\square□

Studdy Solution

STEP 1

What is this asking? Out of 50 patients, what's the chance that 0, 1, 2, 3, 4, *or* 5 patients find the drug ineffective? Watch out! Don't mix up "effective" and "not effective"!Also, "at most 5" includes 5!

STEP 2

1. Find the probability of "not effective".2. Set up the binomial probability formula.3. Calculate the probabilities for 0 to 5 patients.4. Sum the probabilities.

STEP 3

If the drug is effective for 68%68\%68% of patients, that means it's *not* effective for 100%−68%=32%100\% - 68\% = 32\%100%−68%=32% of patients.So the probability of the drug being ineffective for a single patient is 0.320.320.32.Let's call this probability q=0.32q = \textbf{0.32}q=0.32.

STEP 4

STEP 5

We need to calculate P(X=k)P(X=k)P(X=k) for kkk from 0 to 5.Let's do it!

STEP 6

P(X=0)=(500)⋅(0.68)0⋅(0.32)50−0≈2.582249878086908e-20P(X=0) = \binom{50}{0} \cdot (0.68)^0 \cdot (0.32)^{50-0} \approx \textbf{2.582249878086908e-20}P(X=0)=(050​)⋅(0.68)0⋅(0.32)50−0≈2.582249878086908e-20

STEP 7

P(X=1)=(501)⋅(0.68)1⋅(0.32)50−1≈9.399056546095877e-19P(X=1) = \binom{50}{1} \cdot (0.68)^1 \cdot (0.32)^{50-1} \approx \textbf{9.399056546095877e-19}P(X=1)=(150​)⋅(0.68)1⋅(0.32)50−1≈9.399056546095877e-19

STEP 8

P(X=2)=(502)⋅(0.68)2⋅(0.32)50−2≈1.703826764584705e-17P(X=2) = \binom{50}{2} \cdot (0.68)^2 \cdot (0.32)^{50-2} \approx \textbf{1.703826764584705e-17}P(X=2)=(250​)⋅(0.68)2⋅(0.32)50−2≈1.703826764584705e-17

STEP 9

P(X=3)=(503)⋅(0.68)3⋅(0.32)50−3≈2.058053823183734e-16P(X=3) = \binom{50}{3} \cdot (0.68)^3 \cdot (0.32)^{50-3} \approx \textbf{2.058053823183734e-16}P(X=3)=(350​)⋅(0.68)3⋅(0.32)50−3≈2.058053823183734e-16

STEP 10

P(X=4)=(504)⋅(0.68)4⋅(0.32)50−4≈1.862613459045688e-15P(X=4) = \binom{50}{4} \cdot (0.68)^4 \cdot (0.32)^{50-4} \approx \textbf{1.862613459045688e-15}P(X=4)=(450​)⋅(0.68)4⋅(0.32)50−4≈1.862613459045688e-15

STEP 11

P(X=5)=(505)⋅(0.68)5⋅(0.32)50−5≈1.343633738227269e-14P(X=5) = \binom{50}{5} \cdot (0.68)^5 \cdot (0.32)^{50-5} \approx \textbf{1.343633738227269e-14}P(X=5)=(550​)⋅(0.68)5⋅(0.32)50−5≈1.343633738227269e-14

STEP 12

Now, we add up the probabilities we just calculated to find the probability that at most 5 patients find the drug ineffective: P(X≤5)=∑k=05P(X=k)≈1.57864e-14P(X \le 5) = \sum_{k=0}^{5} P(X=k) \approx \textbf{1.57864e-14}P(X≤5)=k=0∑5​P(X=k)≈1.57864e-14

STEP 13

The probability that the drug is not effective for at most 5 patients is approximately **0.0000000000000158**.

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QuestionA new drug has shown to be effective in $68 \%$ of participants during the trials. In a group of 50 patients; what is the probability that the drug is not effective for at most 5 patients?
State answer as a decimal rounded to six decimal places. $\square$

What is this asking? Out of 50 patients, what's the chance that 0, 1, 2, 3, 4, or 5 patients find the drug ineffective? Watch out! Don't mix up "effective" and "not effective"!
Also, "at most 5" includes 5!

1. Find the probability of "not effective".
2. Set up the binomial probability formula.
3. Calculate the probabilities for 0 to 5 patients.
4. Sum the probabilities.

If the drug is effective for $68\%$ of patients, that means it's not effective for $100\% - 68\% = 32\%$ of patients.
So the probability of the drug being ineffective for a single patient is $0.32$ .
Let's call this probability $q = \textbf{0.32}$ .

We need to calculate $P(X=k)$ for $k$ from 0 to 5.
Let's do it!

$P(X=0) = \binom{50}{0} \cdot (0.68)^0 \cdot (0.32)^{50-0} \approx \textbf{2.582249878086908e-20}$

$P(X=1) = \binom{50}{1} \cdot (0.68)^1 \cdot (0.32)^{50-1} \approx \textbf{9.399056546095877e-19}$

$P(X=2) = \binom{50}{2} \cdot (0.68)^2 \cdot (0.32)^{50-2} \approx \textbf{1.703826764584705e-17}$

$P(X=3) = \binom{50}{3} \cdot (0.68)^3 \cdot (0.32)^{50-3} \approx \textbf{2.058053823183734e-16}$

$P(X=4) = \binom{50}{4} \cdot (0.68)^4 \cdot (0.32)^{50-4} \approx \textbf{1.862613459045688e-15}$

$P(X=5) = \binom{50}{5} \cdot (0.68)^5 \cdot (0.32)^{50-5} \approx \textbf{1.343633738227269e-14}$

Now, we add up the probabilities we just calculated to find the probability that at most 5 patients find the drug ineffective: $P(X \le 5) = \sum_{k=0}^{5} P(X=k) \approx \textbf{1.57864e-14}$

The probability that the drug is not effective for at most 5 patients is approximately 0.0000000000000158.