Chapter 7-3 Independent and Dependent Events
Obj: To determine whether an event is dependent or dependent and to determine their probabilities
Why? - Analysts use demographic information and probability to predict results of elections
Independent events – events are independent if the occurrence of one event does not affect the other
Ex –
Probability of Independent Events
𝐼𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑒𝑣𝑒𝑛𝑡𝑠, 𝑡ℎ𝑒𝑛 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)
Ex. 1 Find the Probability of the Following Events:
a) rolling a 6 on one fair die then rolling a 6 on another die
b) tossing heads, then heads, then tails when tossing a fair coin 3 times.
c) tossing a head on a fair coin, then rolling a 6 on a fair die
Dependent events – events are dependent if the occurrence of one event affects the probability of the
other. For example: If there are 3 blue marbles, 4 red, and 2 yellow marble in a bag, what is the
probability of drawing 2 consecutive red marbles if the marbles are not replaced between draws.
P(R, R) =
Conditional Probability P(B|A) – is the probability of event B, given that event A has occurred. This can
be used to find the probability of dependent events
Probability of Dependent Events –
𝐼𝑓 𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑒𝑣𝑒𝑛𝑡𝑠, 𝑡ℎ𝑒𝑛 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴),
𝑤ℎ𝑒𝑟𝑒 𝑃(𝐵|𝐴) 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐵, 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡 𝐴 ℎ𝑎𝑠 𝑜𝑐𝑢𝑟𝑟𝑒𝑑.
B – Explain why the events are dependent. Then find the indicated probability:
The blue die shows a multiple of 3 and the sum is 8
C – Explain why the events are dependent. Then find the indicated probability:
The red die shows a number greater than 4, and the sum is greater than 9
Sometimes we see…Conditional Probability using a table
The table shows domestic migration from 1995 to 2000. A person is
randomly selected. Find each probability:
- That an emigrant is from the West
- That someone selected from the South is an immigrant
- That someone selected is an emigrant and is from the Midwest
As mentioned before, cases involving random selection can be either independent or
dependent. Events are considered independent when there is replacement and
dependent when there is not replacement.
Ex. Two cards are drawn from a deck of 52. Determine if each of the following events is
independent or dependent, then determine the probability:
- Selecting two aces in a row when the first card is replaced.
- Selecting two aces in a row when the first card is NOT replaced.
- Selecting a face card and then a 7 when the first card is NOT replaced.
A bag contains 10 beads – 2 black, 3 white, and 5 red. A bead is selected at random.
Determine if the events are independent or dependent, then find the probability.
- Selecting a white bead, replacing it, then selecting a red bead
- Selecting a white bead, NOT replacing it, then selecting a red bead
- Selecting 3 non-red beads without replacement
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7.3 Homework – pg 503, 2-11, 17-22