If 56% of Americans say they live paycheck to paycheck, find the probability that 7 randomly selected Americans say they live paycheck to paycheck. Please round the final answer to 2 or 3 decimal places.
To find the probability that 7 randomly selected Americans say they live paycheck to paycheck, we'll assume that the probability of an American saying they live paycheck to paycheck is the same for each individual.
Given that 56% of Americans say they live paycheck to paycheck, the probability that any individual says they live paycheck to paycheck is 0.56.
Since we want to find the probability of 7 Americans saying they live paycheck to paycheck, we can use the binomial distribution formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting k successes in n trials,
n is the number of trials (7 in this case),
k is the number of successes we're interested in (7 in this case),
p is the probability of success in a single trial (0.56),
C(n, k) is the number of combinations of n items taken k at a time.
Plugging in the values:
P(X = 7) = C(7, 7) * 0.56^7 * (1-0.56)^(7-7)
Now, let's calculate it step-by-step:
C(7, 7) = 1 (as there is only 1 way to choose all 7 people in this case)
P(X = 7) = 1 * 0.56^7 * (1-0.56)^(7-7)
Simplifying:
P(X = 7) = 0.56^7 * 0.44^0
Since any number raised to the power of 0 is 1:
P(X = 7) = 0.56^7
Calculating:
P(X = 7) ≈ 0.0171
Therefore, the probability that 7 randomly selected Americans say they live paycheck to paycheck is approximately 0.0171 or 1.71%.
To find the probability that 7 randomly selected Americans say they live paycheck to paycheck, we can use the binomial probability formula. The binomial probability formula is:
P(x) = (nCx) * p^x * q^(n-x)
where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of successes
p is the probability of success on a single trial
q is the probability of failure on a single trial
In this case, we have:
n = 7 (the number of trials)
x = 7 (the number of successes, which is when an American says they live paycheck to paycheck)
p = 0.56 (the probability of success, which is the percentage of Americans who say they live paycheck to paycheck)
q = 1 - p = 1 - 0.56 = 0.44 (the probability of failure, which is when an American does not say they live paycheck to paycheck)
Now let's calculate the probability:
P(7) = (7C7) * 0.56^7 * 0.44^(7-7)
= 1 * 0.56^7 * 0.44^0
= 0.56^7
≈ 0.0149 (rounded to 4 decimal places)
Therefore, the probability that 7 randomly selected Americans say they live paycheck to paycheck is approximately 0.0149 (rounded to 2 or 3 decimal places).