a die is thrown.
if even pick 1 mug from bag having 3 black and 4 white mugs.
if odd pick 2 mugs.
what is the probability that the second mug is white
Let's first consider the probability of picking two white mugs from the given back.
Probability of first white mug = White/Total = 4/7
Probability of second white mug = White/New Total = 3/6 = 1/2
So, the probability of picking two white mugs in a row from that bag is (4/7)*(1/2) = 2/7
Now, note that the question asks for the probability that the *second* mug is white, and you only pull a second mug out if you roll an odd number.
So the probability we just found out will be multiplied by (1/2) since an odd number is only rolled half the time.
Actual probability = (2/7)*(1/2) = 1/7
O - odd, E - even, B - black mug , W - white mug
possible outcomes and assuming the mug is not returned if 2 are picked:
E B = (1/2)(3/7) = 3/14
E W = (1/2)(4/7) = 4/14
O WW <--prob--> (1/2)(4/7)(3/6) = 1/7
O WB = (1/2)(4/7)(3/6) = 1/7
O BW <--prob--> (1/2)(3/7)(4/6) = 1/7)
O BB = 1/14
prob(your event) = 1/7+1/7 = 2/7
(note that the sum of the probs of the 6 cases = 1
to clarify the question more
if the outcome of the die is odd, we will pick only one mug so we will not have a second mug at all
sorry
if the outcome of the die is even, we will pick only one mug so we will not have a second mug at all
70% of new employees take a learning lessons.
During first month of work they have probability of 0.04 to make mistakes while those who didn't take lessons (30%) have probability of 0.09.
Given that an employee didn't make mistakes in his first month, what is the probability that he had token lessons
To calculate the probability that the second mug picked is white, we can break down the problem into two cases:
Case 1: The die shows an even number (2, 4, or 6)
In this case, we pick only 1 mug from the bag.
Step 1: Calculate the probability of rolling an even number with a standard die.
Since there are 6 equally likely outcomes when rolling a die (numbers 1 to 6), the probability of getting an even number is 3/6 or 1/2.
Step 2: Calculate the probability of picking a black mug from the bag.
The bag has a total of 3 black mugs and 4 white mugs. So the probability of picking a black mug is 3/7.
Step 3: Calculate the probability of picking a white mug from the bag.
Since there is only one mug picked in this case, if the mug picked is not black, it must be white. So the probability of picking a white mug in this case is 1 - (3/7) = 4/7.
Case 2: The die shows an odd number (1, 3, or 5)
In this case, we pick 2 mugs from the bag.
Step 1: Calculate the probability of rolling an odd number with a standard die.
Similar to Case 1, the probability of rolling an odd number is also 1/2.
Step 2: Calculate the probability of picking 2 black mugs from the bag.
If we pick 2 mugs, then we need to calculate the probability of picking a black mug twice in a row.
The probability of picking the first black mug is 3/7.
After picking the first black mug, there would be 2 black mugs left out of the remaining 6 mugs. So, the probability of picking a black mug again is 2/6 = 1/3.
To find the probability of picking black mugs in two consecutive events, we multiply the probabilities: (3/7) * (1/3) = 1/7.
Step 3: Calculate the probability of picking at least 1 white mug from the bag.
Since we are interested in the probability of picking the second mug as white, we will calculate the probability of picking 2 black mugs and subtract it from 1.
Probability of picking at least 1 white mug = 1 - (1/7) = 6/7.
Now let's calculate the overall probability by considering both cases:
Probability of picking the second mug as white =
(Probability of an even number) * (Probability of picking a white mug in Case 1) +
(Probability of an odd number) * (Probability of picking at least 1 white mug in Case 2) =
(1/2) * (4/7) + (1/2) * (6/7) =
4/14 + 6/14 =
10/14 =
5/7.
Therefore, the probability that the second mug picked is white is 5/7.