How I look at it is that since the number of throws is lesser, there is lesser chance of all numbers coming up. Hence, 4 has a chance of coming up more than 1/4 times in two throws.
Another way to look at it is that in two throws only two numbers can come up and the chance of 4 coming up once in two trials will be higher than 1/6
4 not 4
not 4 4
1/6*5/6+5/6*1/6 = 25/36>1/4
Let n be the number of times the dice is rolled to ensure that the probability of a 4 is at least 1/4.
n*(probability of rolling a 4 in one turn)>= 1/4
Hence "minimum number of times to ensure that the probability that a 4 is rolled is at least 1/4" >= 1.5 = 2 (since we cannot have 1.5 rolls)
By this logic, in 6 throws, you have a probability of getting 4 as 1. I doubt if your approach is correct.
And in 7 and above throws the probability to get 4 will be more than 1. I dont know if I am missing something
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