Probability of coin flip and rolling a dice

Question

Li Yan flips two coins and rolls a standard six-sided die. What is the probability that Li Yan does not both roll an even number on the die, and throw 2 heads with the coins? 
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 here i thought i could take the approach of

not rolling an even number = 3/6 = 1/2 
 not throwing 2 heads = 3/4

so if we multiply the two we would get 3/8

however the solution says that it is 7/8

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this is the solution 
 Since the question is asking for the probability that Li Yan does    not    both roll an even number and throw two heads, the fastest way to solve this problem is to find the probability that Li Yan    does    roll an even number and get 2 heads, and subtract that probability from 1. 
 The probability of Li Yan rolling an even number is = 3/6 = 1/2

The probability of Li Yan getting 2 heads on 2 flips is = 1/4 
 Thus, the probability of Li Yan rolling an even number and getting two heads is:1/8

and thus the complement of what we want is 7/8

i'm trying to understand what is wrong with my approach

MartyMurray · Answer

You calculated the probability that he won't do either.

In other words, you calculated the probability of not 2 heads AND not an even number.

The correct probability is higher because he can do one OR the other and still not do both.

vangoghskindle · Answer

Coin flips: 2 options per coin, so 2 * 2 = 4 total outcomes (HH, HT, TH, TT)
 Die roll: 6 options, so 6 total outcomes (1, 2, 3, 4, 5, 6)
 Total combined outcomes: 4 * 6 = 24

Step 2: Favorable Outcomes for the Desired Event (Both Heads and Even Roll)

Favorable coin flip outcomes: 1 (HH)
 Favorable die roll outcomes: 3 (2, 4, 6)
 Total favorable outcomes for the desired event: 1 * 3 = 3

Step 3: Favorable Outcomes for the Complement Event (Not Both Heads and Not Even Roll)

Total outcomes - Favorable outcomes for the desired event = 24 - 3 = 21

Step 4: Probability of the Complement Event

Probability = Favorable outcomes for the complement event / Total outcomes
 Probability = 21 / 24 = 7/8

Therefore, the probability of not getting both two heads on the coins and an even number on the die is 7/8.

lavanya.18 · Answer

Let event 'A' be the event that Li Yan rolls an even number on the die.
Let event 'B' be the event that Li Yan flips 2 heads with the coins.
So, to find P(A)= 3/6=1/2
And, P (B)= 1/4
Both P(A) and P(B) are independent, so the probability of both happening together is their product: 1/2*1/4=1/8
We are asked to find the probability that Li Yan does not both roll an even number and flip 2 heads. The probability of this complement event is: 1- 1/8= 7/8 ->ANSWER

Probability of coin flip and rolling a dice

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