At a large University, 60% of the students have a Facebook page that they've checked today. 12% of the students have an Instagram page that they've checked today. Included in those percentages is the 3% of students who have checked both a Facebook page and a Instagram page today. If a student from this University is chosen at random, find the probability that that student...
a) has a Facebook page or an Instagram page that they've checked today.
b) has an Instagram page that they've checked today, given that have a Facebook page that they've checked today.
c) has neither a Facebook page nor an Instagram page that they've checked today.
d) Is the event F, that the randomly selected student has a Facebook page that they've checked today, independent of the event I- that the randomly selected student has an Instagram page that they've checked today? EXPLAIN how you can reach your conclusion.
e) Is the event F, that the randomly selected student has a Facebook page that they've checked today, mutually exclusive of the event I-, that the randomly selected student has a Instagram page that they've checked today? EXPLAIN how you can reach your conclusion.
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Let's define F as the event that the randomly selected student has a Facebook page that they've checked today and I as the event that the randomly selected student has an Instagram page that they've checked today.
a) To find the probability that a student has a Facebook page or an Instagram page that they've checked today, we can use the principle of inclusion-exclusion. The formula is P(F U I) = P(F) + P(I) - P(F ∩ I), where P(F) is the probability of having a Facebook page, P(I) is the probability of having an Instagram page, and P(F ∩ I) is the probability of having both.
P(F) = 60% = 0.6
P(I) = 12% = 0.12
P(F ∩ I) = 3% = 0.03
P(F U I) = 0.6 + 0.12 - 0.03 = 0.69
Therefore, the probability that a randomly selected student has a Facebook page or an Instagram page that they've checked today is 0.69.
b) To find the probability that a student has an Instagram page that they've checked today, given that they have a Facebook page that they've checked today, we use the conditional probability formula. P(I|F) = P(F ∩ I) / P(F)
P(F ∩ I) = 0.03 (as given)
P(F) = 0.6 (as given)
P(I|F) = 0.03 / 0.6 = 0.05
Therefore, the probability that a randomly selected student has an Instagram page that they've checked today, given that they have a Facebook page that they've checked today is 0.05.
c) To find the probability that a student has neither a Facebook page nor an Instagram page that they've checked today, we subtract the probability of having either a Facebook or Instagram page from 1. P(neither) = 1 - P(F U I)
P(F U I) = 0.69 (as calculated in part a)
P(neither) = 1 - 0.69 = 0.31
Therefore, the probability that a randomly selected student has neither a Facebook page nor an Instagram page that they've checked today is 0.31.
d) To determine whether the event F (having a Facebook page) is independent of the event I (having an Instagram page), we need to compare the joint probability of both events occurring (P(F ∩ I)) with the product of their individual probabilities (P(F) * P(I)).
If P(F ∩ I) = P(F) * P(I), then the events are independent.
P(F ∩ I) = 0.03 (as given)
P(F) = 0.6 (as given)
P(I) = 0.12 (as given)
P(F) * P(I) = 0.6 * 0.12 = 0.072
Since P(F ∩ I) ≠ P(F) * P(I), the events F and I are not independent.
e) Two events are mutually exclusive if they cannot occur simultaneously, meaning P(F ∩ I) = 0. In this case, we already know that P(F ∩ I) = 0.03 ≠ 0. Therefore, the events F and I are not mutually exclusive.
In conclusion, the events F and I are neither independent nor mutually exclusive.
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