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On his drive to work, Leo listens to one of three radio stations A, B or C. He first turns to A. If A is playing a song he likes, he listens to it; if not, he turns it to B. If B is playing a song he likes, he listens to it; if not, he turns it to C. If C is playing a song he likes, he listens to it; if not, he turns off the radio. For each station, the probability is \(0.30\) that at any given moment the station is playing a song Leo likes. On his drive to work, what is the probability that Leo will hear a song he likes?
Let's understand what Leo does step by step. Leo has a very specific routine:
The key insight is that Leo stops as soon as he finds something he likes. We want to find the probability that he ends up hearing a song he likes (rather than turning off the radio).
For each station, there's a \(\mathrm{0.30}\) probability that Leo likes what's playing, which means there's a \(\mathrm{0.70}\) probability that he doesn't like what's playing.
Process Skill: TRANSLATE - Converting the sequential decision process into clear probability scenarios
Let's map out all the ways Leo could end up hearing a song he likes:
Path 1: Leo likes station A
Path 2: Leo doesn't like A, but likes station B
Path 3: Leo doesn't like A or B, but likes station C
Path 4: Leo doesn't like any station
We want the probability of Paths 1, 2, or 3 happening (any path where he actually hears a song).
Now let's calculate the probability of each path where Leo hears a song:
Path 1 - Likes station A:
Probability = \(\mathrm{0.30}\)
(Simply the chance he likes the first station he tries)
Path 2 - Doesn't like A, likes B:
Probability = (doesn't like A) × (likes B)
Probability = \(\mathrm{0.70} \times \mathrm{0.30} = \mathrm{0.21}\)
Path 3 - Doesn't like A or B, likes C:
Probability = (doesn't like A) × (doesn't like B) × (likes C)
Probability = \(\mathrm{0.70} \times \mathrm{0.70} \times \mathrm{0.30} = \mathrm{0.147}\)
These are the only three ways Leo ends up hearing a song he likes.
To find the total probability that Leo hears a song he likes, we add up all the successful paths:
Total probability = Path 1 + Path 2 + Path 3
Total probability = \(\mathrm{0.30} + \mathrm{0.21} + \mathrm{0.147}\)
Total probability = \(\mathrm{0.657}\)
Let's verify this makes sense: we can also calculate this as 1 minus the probability he doesn't hear any song he likes.
Probability of hearing no song = (doesn't like A) × (doesn't like B) × (doesn't like C)
Probability of hearing no song = \(\mathrm{0.70} \times \mathrm{0.70} \times \mathrm{0.70} = \mathrm{0.343}\)
So probability of hearing a song = \(\mathrm{1} - \mathrm{0.343} = \mathrm{0.657}\) ✓
The probability that Leo will hear a song he likes is \(\mathrm{0.657}\).
This matches answer choice D. 0.657
1. Misunderstanding the sequential nature of Leo's decision process
Students often miss that Leo stops as soon as he finds a song he likes, instead thinking he checks all three stations regardless. This leads to incorrectly calculating probabilities for scenarios where Leo continues searching after already finding a song he likes on an earlier station.
2. Confusing "probability of hearing a song he likes" with "probability of liking each individual station"
Some students mistakenly think they need to find the probability that Leo likes songs on all stations, or focus only on one station, rather than understanding that we want the probability of ANY successful outcome across the sequential process.
3. Not recognizing this as a sequential probability problem with mutually exclusive paths
Students may try to use simple probability rules for independent events rather than mapping out the distinct sequential paths (Path 1: likes A, Path 2: doesn't like A but likes B, Path 3: doesn't like A or B but likes C).
1. Using 0.30 instead of 0.70 for "doesn't like" probabilities
When calculating Path 2 and Path 3, students often forget that if the probability of liking a station is \(\mathrm{0.30}\), then the probability of NOT liking it is \(\mathrm{0.70}\). They might incorrectly use \(\mathrm{0.30}\) for both scenarios.
2. Arithmetic errors when multiplying probabilities for sequential events
For Path 3, students need to calculate \(\mathrm{0.70} \times \mathrm{0.70} \times \mathrm{0.30} = \mathrm{0.147}\). Common mistakes include getting \(\mathrm{0.49} \times \mathrm{0.30} = \mathrm{0.147}\) wrong, or making errors in the earlier calculations like \(\mathrm{0.70} \times \mathrm{0.30}\) for Path 2.
3. Incorrectly adding probabilities or forgetting to include all successful paths
Students might forget to add all three successful paths together, or make addition errors when computing \(\mathrm{0.30} + \mathrm{0.21} + \mathrm{0.147} = \mathrm{0.657}\).
1. Selecting the probability of the complementary event (not hearing any song)
Since the solution shows that \(\mathrm{P}(\text{not hearing any song}) = \mathrm{0.343}\), students might accidentally select an answer choice close to this value instead of \(\mathrm{1} - \mathrm{0.343} = \mathrm{0.657}\).
2. Choosing a probability that represents only one successful path instead of all paths combined
Students might select \(\mathrm{0.30}\) (only Path 1) or other individual path probabilities rather than the sum of all successful outcomes, especially if answer choices include these intermediate values.