A spinner is divided into 8 equal sections. Five sections are red, and three are green. If the spinner is spun three times, what is the probability that it lands on red exactly twice?
Answer (3)
(A) green and grey ;
The probability that the spinner lands on red exactly twice in 3 spins is 512 225 .
The probability of the spinner landing on red exactly twice in 3 spins is given by the binomial probability formula:
P ( X = k ) = ( k n ) ⋅ p k ⋅ ( 1 − p ) n − k
where n is the number of trials, k is the number of successes, is the probability of success on a single trial, and ( k n ) is the binomial coefficient representing the number of ways to choose k successes from n trials.
In this case, n = 3 (since the spinner is spun 3 times), k = 2 (we want exactly 2 red outcomes), and p = 8 5 (the probability of landing on red in a single spin, as there are 5 red sections out of 8 total sections).
First, we calculate the probability of landing on red in a single spin:
p = Total number of sections Number of red sections = 8 5
The probability of not landing on red (i.e., landing on green) in a single spin is:
1 − p = 1 − 8 5 = 8 3
Now, we use the binomial probability formula to find the probability of landing on red exactly twice in 3 spins:
P ( X = 2 ) = ( 2 3 ) ⋅ ( 8 5 ) 2 ⋅ ( 8 3 ) 3 − 2
P ( X = 2 ) = 3 ⋅ ( 8 5 ) 2 ⋅ ( 8 3 )
P ( X = 2 ) = 3 ⋅ 64 25 ⋅ 8 3
P ( X = 2 ) = 3 ⋅ 512 75
P ( X = 2 ) = 512 225
Therefore, the probability that the spinner lands on red exactly twice in 3 spins is 512 225 .
The probability that the spinner lands on red exactly twice in three spins is 512 225 . This is calculated using the binomial probability formula, with values derived from the spinner's configuration. The probability of landing on red is 8 5 and not red is 8 3 .
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