It looks like the image you provided contains a problem related to probability and hypothesis testing. I'll break down the questions for you.

The task involves 500 tosses of a coin, and a random variable $X$ represents the number of heads (denoted as "plat" or "heads" in the image) observed in those tosses. The main points are:

a) Determining Probabilities:

You are asked to find the probabilities $P(X \leq 230)$ and $P(X \geq 270)$, assuming that the coin is symmetric. This means that the probability of heads is 0.5. Since this follows a binomial distribution, we can approximate it using a normal distribution for large $n$.

b) Formulating Hypotheses for a Test:

Here, you are asked to formulate a null hypothesis for a two-tailed test at a 5% significance level, to determine whether the coin is symmetric. A standard approach would be testing if the observed number of heads deviates significantly from what would be expected for a fair coin.

c) Critical Value for the Test:

Finally, you are asked to find the critical value for the test, and use that to determine whether the null hypothesis can be rejected, based on the given number of heads (270 out of 500 tosses).

Let me go through each part in more detail. Would you like to proceed with part a first?