Answer:
P = 0.545
Step-by-step explanation:
Here we can find the probability Q, in which you need one or two spins to win the game, and then:
P = 1 - Q
Will be the probability that it takes 3 or more spins to win.
Case where you need only one spin:
There are 3 out of 12 possible outcomes, then the probability of winning with only one spin is:
p = 3/12.
Case where you need two spins to win:
Here we should get a normal sushi roll in the first spin and a wasabi bomb in the second.
The probability of getting a sushi roll in the first spin is equal to the quotient between the number of sushi rolls (9) and the total number of possible outcomes (12), then:
p1 = 9/12
For the second spin we need to get a wasabi bomb, assuming that the susi roll is not replaced (so now there are 8 sections with sushi rolls and 3 with wasabi bombs, if you get the empty section you roll again or something like that)
Now the probability of getting a wasabi bomb is equal to the quotient between the number of wasabi bombs remaining (3) and the total number of outcomes remaining (11), this is:
p2 = 3/11
The joint probability is the product of the two individual probabilities:
p = p1*p2 = (9/12)*(3/11)
Now, the total probability of winning with one or two spins is equal to the sum of the probabilities for each case, then:
Q = (9/12)*(3/11) + 3/12 = 0.455
Then the probability of needing 3 or more spins to win is:
P = 1 - Q = 1 - 0.455 = 0.545