According to a survey, 21% of motorists in a large city are uninsured. Suppose that currently 21% of all motorists in this city are uninsured. Two motorists are selected at random from this city. Let x denote the number of motorists in this sample of two who are uninsured. Construct the probability distribution table of x.
Ah, probability distributions, the mathematics of surprises! Let's dive into constructing the probability distribution table for x.
To do this, we need to consider the possible values x can take on, which are 0, 1, and 2. Now, let's calculate the probabilities associated with each value.
When x = 0, it means both selected motorists are insured. The probability of this happening can be calculated as:
P(x = 0) = (probability of first motorist being insured) * (probability of second motorist being insured)
= 0.79 * 0.79 (since 79% of motorists are insured)
= 0.6241
When x = 1, one of the motorists is uninsured and the other is insured. There are two possible scenarios for this:
1. First motorist is uninsured, and the second one is insured.
P(x = 1, scenario 1) = (probability of first motorist being uninsured) * (probability of second motorist being insured)
= 0.21 * 0.79
= 0.1659
2. First motorist is insured, and the second one is uninsured.
P(x = 1, scenario 2) = (probability of first motorist being insured) * (probability of second motorist being uninsured)
= 0.79 * 0.21
= 0.1659
To find the total probability of x = 1, we sum the probabilities of both scenarios:
P(x = 1) = P(x = 1, scenario 1) + P(x = 1, scenario 2)
= 0.1659 + 0.1659
= 0.3318
Lastly, when x = 2, it means both selected motorists are uninsured. The probability can be calculated as:
P(x = 2) = (probability of first motorist being uninsured) * (probability of second motorist being uninsured)
= 0.21 * 0.21
= 0.0441
Now, putting it all together, we have our probability distribution table:
x | 0 | 1 | 2
-------------------------
P(x) | 0.6241| 0.3318| 0.0441
Remember, probabilities can be as unpredictable as a box of chocolates, but this distribution table should help you understand the likelihood of each scenario.
To construct the probability distribution table of x, we need to calculate the probabilities of each possible outcome.
We are given that currently 21% of all motorists in this city are uninsured. Therefore, the probability that a randomly selected motorist is uninsured is 0.21, and the probability that a randomly selected motorist is insured is 1 - 0.21 = 0.79.
Let's calculate the probabilities for each possible value of x:
When x = 0 (both motorists are insured):
P(x = 0) = P(both insured) = P(insured) * P(insured) = 0.79 * 0.79 = 0.6241
When x = 1 (one motorist is insured and one is uninsured):
P(x = 1) = P(insured) * P(uninsured) + P(uninsured) * P(insured) = 0.79 * 0.21 + 0.21 * 0.79 = 0.3321 + 0.3321 = 0.6642
When x = 2 (both motorists are uninsured):
P(x = 2) = P(both uninsured) = P(uninsured) * P(uninsured) = 0.21 * 0.21 = 0.0441
Finally, we can construct the probability distribution table as follows:
x | P(x)
--------------
0 | 0.6241
1 | 0.6642
2 | 0.0441
This table represents the probability distribution of x, the number of motorists in a sample of two who are uninsured in the city.
To construct the probability distribution table of x, we need to calculate the probabilities associated with each possible outcome for x.
Since there are two motorists selected at random, x can take on the values 0, 1, or 2. Let's calculate the probability for each value:
1. When x = 0 (both motorists are insured):
The probability of the first motorist being insured is 79% (100% - 21%), and the probability of the second motorist being insured is also 79%. To find the probability of both events happening, we multiply the probabilities together: 0.79 * 0.79 = 0.6241.
2. When x = 1 (one motorist is uninsured):
We have two possible scenarios: the first motorist being uninsured and the second motorist being insured, or the first motorist being insured and the second motorist being uninsured. Both scenarios have the same probability since order does not matter. The probability for each scenario is: 0.21 * 0.79 = 0.1659. To find the total probability, we add the probabilities of both scenarios: 0.1659 + 0.1659 = 0.3318.
3. When x = 2 (both motorists are uninsured):
The probability of the first motorist being uninsured is 21%, and the probability of the second motorist being uninsured, given that the first motorist is uninsured, is also 21%. To find the probability of both events happening, we multiply the probabilities together: 0.21 * 0.21 = 0.0441.
Now, we can construct the probability distribution table of x:
x | Probability
--------------
0 | 0.6241
1 | 0.3318
2 | 0.0441
Make sure that the sum of all probabilities is equal to 1, which is the case here: 0.6241 + 0.3318 + 0.0441 = 1.