Probability Calculation for 10 Coin Tosses
We need to find the probability that all 10 outcomes are the same when an unbiased coin is tossed 10 times.
Identifying Favorable and Total Outcomes
- Total Possible Outcomes: For each toss, there are 2 possible outcomes (Heads or Tails). Since the coin is tossed 10 times, the total number of possible sequences is $2^{10}$. $ \text{Total Outcomes} = 2^{10} = 1024 $
- Favorable Outcomes: The event "all outcomes are the same" means either all 10 tosses are Heads (HHHHHHHHHH) or all 10 tosses are Tails (TTTTTTTTTT). $ \text{Favorable Outcomes} = 2 $
Calculating the Probability
The probability of an event is calculated as the ratio of favorable outcomes to total possible outcomes. $ P(\text{All Same}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} $ $ P(\text{All Same}) = \frac{2}{1024} = \frac{1}{512} $
Expressing Probability in Scientific Notation
The question asks for the probability in the format $P \times 10^{-5}$. First, convert the fraction to a decimal: $ \frac{1}{512} \approx 0.001953125 $ Now, we need to express this decimal in the form $P \times 10^{-5}$: $ 0.001953125 = P \times 10^{-5} $ To find $P$, multiply the decimal by $10^5$: $ P = 0.001953125 \times 10^5 $ $ P = 195.3125 $
The probability is approximately $195.3125 \times 10^{-5}$. The value $195.3125$ falls within the range specified (between 191 and 199).
Important Questions from Population Genetics and Epigenetics
Which of the following conditions will contribute to the stability of a gene pool in a natural population?
P. Large population
Q. No net mutation
R. Non-random mating
S. No selection
- In a random mating population, $Y$ and $y$ are dominant and recessive alleles, respectively. If the frequency of $Y$ allele in both sperm and egg is $0.70$, then the frequency of $Y/y$ heterozygotes (rounded off to two decimal places) is ________
- An isolated population on an island has the following genotypic frequencies:Assuming that there are only two alleles (A and a) for the gene, the genotypic frequency of AA in the next generation will be ________
Genotype AA Aa aa Frequency 0.3 0.4 0.3 In a genetic study, $80$ people were found to have alleles for polydactyly. Only $36$ of them were polydactylous. What is the extent of penetrance percentage? _______________
- Consider a population of $10,000$ individuals, of which $2500$ are homozygotes (PP) and $3000$ are heterozygotes (Pp) genotype. The frequency of allele p in the population is _________.
